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Preheat the oven to 375 degrees F.
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Lay out the bread slices.
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Divide the ham among them, making sure it doesn't extend over the edges of the bread.
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Place the cheese over the ham.
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If the cheese is larger than the bread, bend it over to fit.
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Heat 2 large ovenproof nonstick pans or griddles over medium heat.
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(If you have only 1 large pan, make 2 sandwiches and keep them warm in the oven while you make the second batch.)
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Add 1 tablespoon of the butter to each pan.
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When it has melted, add half the bread, cheese-side up to each pan and cook for 1 to 2 minutes, or until the bottoms are golden brown.
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Transfer the pans to the oven for 2 to 3 minutes to melt the cheese.
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Meanwhile, melt the remaining 1 tablespoon of butter in a large ovenproof skillet and fry the eggs.
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Cook the eggs until the bottoms are set, then place the skillet in the oven for a minute to set the top of the whites.
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(We cook the eggs in 4 to 5-inch individual skillets.)
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When the cheese is melted, remove the sandwiches from the oven.
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Place 2 slices together to make each sandwich and put each sandwich on a serving plate.
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Place an egg on top of each sandwich.
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Pour about 1/4 cup of the sauce over the white of each egg, leaving the yolk uncovered.
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Grind black pepper over each egg and garnish the eggs with a diagonal sprinkling of chopped parsley.
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Serve with frites, if desired.
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Jean-Louis Palladin was a close friend and one of the greatest chefs I've ever known.
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And he made some of the best brioche I've ever tasted.
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This is his recipe.
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Start it a day before you want to make it, as the dough has to rest overnight.
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Combine the water and yeast in a small bowl.
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Let stand for 10 minutes, then stir until the yeast is completely dissolved.
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Set aside.
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Sift together the flours, sugar, and salt into the bowl of a stand mixer fitted with the dough hook.
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Add the eggs and beat for 1 minute at low speed, scraping down the sides of the bowl with a rubber spatula as needed.
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Slowly add the dissolved yeast and continue beating at low speed for 5 minutes.
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Stop the machine, scrape any dough off the hook, and beat for another 5 minutes.
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Add about one-quarter of the butter cubes at a time, beating for about 1 minute after each addition.
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Once all the butter has been added, beat for 10 minutes more.
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Place the dough in a large floured bowl and cover with plastic wrap.
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Set aside in a warm place until doubled in size, about 3 hours.
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Turn the dough out onto a generously floured work surface and gently work the air bubbles out by folding the dough over several times while lightly pressing down on it.
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Return the dough to the bowl, cover with plastic wrap, and refrigerate overnight.
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The dough is now ready to shape or use in another recipe.
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Generously butter two 8 1/2 by 4 1/2-inch loaf pans.
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Turn the dough out onto a floured work surface.
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With floured hands, divide the dough in half and shape it into two rectangles that fit in the loaf pans.
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Place the dough in the pans.
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Let the dough rise uncovered in a warm place until it is about 1/2-inch above the top of the pans, about 3 hours.
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Preheat the oven to 350 degrees F.
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Bake the brioche in the center of the oven until it is well browned on top and sounds hollow when tapped on the bottom, 35 to 40 minutes.
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Remove the brioche from the oven and immediately turn out onto a wire rack.
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If serving immediately, let the breads cool for 10 minutes, then slice.
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If serving within a few hours, wrap the hot bread in aluminum foil and set aside at room temperature until ready to use.
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To freeze, wrap the hot bread in foil and promptly freeze.
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The bread can be kept frozen for up to 1 month; when ready to use, reheat (without thawing and still wrapped in the foil) in a 250 degree F oven until heated through, 20 to 25 minutes.
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If using the brioche for croutons, let sit at room temperature uncovered to dry for a day.
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Mornay, a roux-thickened white sauce, is a luxurious cheese sauce that we use for gratineed scallops, macaroni and cheese, croque-monsieurs and -madames, and crepes.
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Melt the butter in a medium heavy saucepan set on a diffuser over medium heat.
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Add the onion and a pinch of salt and cook slowly, stirring occasionally, for 2 to 3 minutes, or until the onion is translucent.
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Sprinkle in the flour and cook for about 3 minutes, stirring constantly so that the roux doesn't burn or color.
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Whisking constantly, add the milk and cream and whisk until fully incorporated.
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Bring to a simmer, whisking, then add the bay leaf, peppercorns, and cloves.
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Move the pan to 1 side of the diffuser, away from direct heat to avoid scorching, and bring back to a gentle simmer.
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Adjust the heat to maintain a gentle simmer and cook, whisking occasionally, reaching into the corners of the pan, for about 30 minutes.
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(If the sauce does begin to scorch, pour it into a clean pan--don't scrape the bottom of the pan--and continue.)
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Remove the sauce from the heat and season to taste with salt, a grating of nutmeg, and a pinch of white pepper.
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Strain the sauce, add the cheese, and whisk to melt.
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Use immediately, or place in a storage container, press a piece of plastic wrap against the surface to keep a skin from forming, and refrigerate for up to a week.
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If the sauce is too thick after refrigeration, it can be thinned with a little heavy cream.
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VARIATION: For 3 cups sauce, use 4 tablespoons (2 ounces) butter, 2/3 cup diced onion, 1/4 cup flour, 22/3 cups milk, 11/3 cups cream, 4 peppercorns, 4 cloves, and 1/2 cup cheese.
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Make the sauce in a large saucepan.
