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1
Preheat oven to 400 degrees F (200 degrees C). Grease a 1-quart baking dish with 1 teaspoon butter.
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2
Arrange bread slices in a single layer in the baking dish.
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3
Whisk milk, 1 egg, nutmeg, salt, and pepper together in a bowl. Pour over bread slices. Sprinkle Parmesan cheese on top. Let stand until milk mixture is absorbed, 15 to 30 minutes. Dot remaining 2 teaspoons butter over Parmesan cheese.
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4
Toss asparagus with olive oil in a bowl. Spread on a baking sheet. Season with salt and pepper.
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5
Bake bread slices in the preheated oven until puffed up, about 10 minutes. Place asparagus in the oven. Continue baking bread slices until golden brown, about 10 minutes more.
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6
Fill a large saucepan with 2 to 3 inches of water and bring to a boil. Reduce the heat to medium-low to bring water to a gentle simmer.
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7
Crack 1 egg into a small bowl then gently slip egg into the simmering water, holding the bowl just above the surface of water. Repeat with the remaining egg. Cook eggs until the whites are firm and the yolks have thickened but still runny, 2 1/2 to 3 minutes. Remove the eggs from the water with a slotted spoon.
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8
Divide bread between 2 plates. Top each portion with a poached egg. Season with salt and pepper. Serve asparagus on the side.
