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1
In a small saucepan over medium heat, melt 1 Tbs. of the butter. Whisk in the flour and cook, whisking constantly, until the mixture is pale golden, about 3 minutes. Slowly add the milk, whisking constantly. Simmer, continuing to whisk, until the sauce is smooth and thickened, 6 to 8 minutes. Stir in the 1/2 tsp. salt plus black pepper and cayenne, to taste. Set the bechamel sauce aside.
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2
Preheat a panini press to 375u00b0F or medium according to the manufacturer's instructions.
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3
Brush one side of each bread slice with oil. Place the slices, oiled side down, on a clean work surface. Spread the mustard on 4 of the slices and top each with 2 slices of ham, folding the ham if necessary to keep it even with the edges of the bread. Spread the bechamel sauce evenly over the ham and sprinkle the cheese on top, dividing evenly. Top each with one of the remaining bread slices, oiled side up.
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4
Place the sandwiches on the preheated panini press and cook according to the manufacturer's instructions until golden and crispy, about 5 minutes.
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5
Meanwhile, in a large fry pan over medium-high heat, melt the remaining 1 Tbs. butter. When the butter foams, break the eggs into the pan, spacing them about 1 inch apart. If using egg rings, spray the inside of 4 rings with nonstick cooking spray, place in the pan and break an egg into each ring. Season the eggs with salt and black pepper. Cook until the whites are firm, about 3 minutes. Remove the egg rings, if using. Flip the eggs over and continue cooking until the whites are cooked through but the yolks are still runny, about 1 minute more.
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6
Slide an egg onto each sandwich, sprinkle with parsley and serve immediately. Serves 4.
