Plus: More Brunch Ideas
In large ovenproof nonstick skillet, heat the olive oil. Add the prosciutto strips and cook over moderately high heat until crisp, about 7 minutes. Transfer the prosciutto to a paper towel-lined plate. Add the asparagus to the rendered prosciutto fat in the skillet and cook until crisp-tender and beginning to brown, about 5 minutes. Season the asparagus with salt and pepper and add them to the prosciutto.
Using a 1-inch round cookie cutter, punch a hole in the center of each slice of bread. Set the bread slices in the skillet and move them around so they can soak up the remaining fat. Crack an egg into each hole and cook over moderately high heat for 2 minutes. Using a thin spatula, carefully flip each slice of bread and cook for 2 minutes more.
Turn on the broiler. Scatter the prosciutto and asparagus over the bread and top with the grated cheese. Broil 6 inches from the heat for 1 minute, until the cheese is bubbling. Transfer to plates and serve.