In a large, nonstick skillet, cook the bacon over moderate heat, turning once, until crisp, about 4 minutes per side. Drain on paper towels.
Wipe out the skillet and melt the butter in 1 tablespoon of olive oil. Add the mushrooms, season with salt and pepper and cover. Cook over moderate heat until the mushrooms soften and release their liquid, about 7 minutes. Uncover and continue to cook over moderate heat, stirring often, until the mushrooms are richly browned, about 8 minutes longer. Transfer the mushrooms to a bowl.
Wipe out the skillet and coat it with a thin film of oil. Fry the eggs over-easy over moderate heat, about 3 minutes on the first side and 30 seconds to 1 minute on the second side. Season the eggs with salt and pepper.
On a work surface, spread 4 slices of the toast with mayonnaise and top with the lettuce, tomato slices and bacon. Cover with 4 slices of toast, spread them each with mayonnaise and top with the mushrooms and fried eggs. Cover the sandwiches with the remaining toast, cut them in half and serve.