How to Make It
Melt 1 tablespoon of the butter in a large skillet. Add the shallots and cook over moderate heat, stirring, until wilted, about 3 minutes. Raise the heat to high and add the mushrooms and lemon juice. Cook, stirring occasionally, until the mushrooms release their liquid. Season with salt and pepper and cook until the liquid has evaporated and the mushrooms start to brown, about 4 minutes. Add the paprika and cook, stirring, until fragrant, about 2 minutes. Transfer the mushrooms to a bowl.
Wipe out the skillet and melt 1 more tablespoon of the butter. Arrange 2 slices of hollowed-out bread in the skillet and break an egg into each hole. Cook over low heat until the bread is well-browned on the bottom and the yolks are partially set, about 3 minutes. Turn the bread over and cook the second sides until the yolks are barely set, about 3 more minutes. Turn the bread again, set a slice of cheese on each toast and season with salt and pepper.
Spread 1 heaping teaspoon of salsa on 1 of the toasts and cover with 1/3 cup of the mushrooms; invert the other toast on top, cheese side down. Transfer the sandwich to a plate and repeat with the remaining ingredients to make 3 more bull's eyes.