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To poach the eggs, in a pot on the stovetop, bring the water and vinegar or lemon juice to a simmer.
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(You need 1 tablespoon of vinegar per cup of water.
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Also, if you use a nonstick pot, the water should be 1 inch deep.
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If you are using a regular pot, it needs to be about 3 inches deep.
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The vinegar or lemon juice will help the eggs keep their shape and not spread in water.)
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Meanwhile, place a skillet on medium heat.
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Spread the butter on both sides of the bread slices.
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Toast bread in the skillet on both sides.
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Place the slices of cheese on top of the bread, turn off the heat, cover, and let the cheese melt.
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At this point, the water in pot should be simmering.
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Carefully break the eggs one at a time into a small bowl, making sure the yolk is intact and the egg is good.
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Gently drop the egg in the simmering water.
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Do the same with the remaining egg.
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Turn off the heat, cover, and let the eggs cook for 4-5 minutes.
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This will make the egg yolk barely runny.
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If you want them softer, cook them for less time.
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Also, if you cook more eggs at once, you have to keep them longer, about 7-8 minutes for 4 eggs, for example, as they absorb more heat and need to cook longer to get the same consistency.
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Very carefully spoon out the eggs and place on a paper towel for a few seconds to dry them.
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Mix the mayonnaise and mustard in a bowl.
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Set aside.
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Place the slices of smoked salmon on top of the melted cheese.
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Add 1 poached egg on top of each slice.
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I let the bread sit in the skillet up till before serving, so they stay warm.
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Place the bread slices on a plate.
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Spoon over some mayo-mustard sauce.
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Season with little salt (not too much as the cheese and salmon are already salty), fresh ground pepper, some cayenne and/or a little grated nutmeg.
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Enjoy warm, with a side of salad, or just as it is.