1 tablespoon extra-virgin olive oil, plus more for frying
1 pound white mushrooms, thinly sliced
Salt and freshly ground pepper
4 large eggs
Twelve 1/4-inch thick slices of whole wheat sandwich bread, toasted
4 romaine lettuce leaves
8 thin tomato slices
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.
In the skillet, melt the butter in the 1 tablespoon of olive oil. Add the mushrooms and season with salt and pepper. Cover and cook over moderate heat until the mushrooms soften and release their liquid, about 7 minutes. Uncover and cook, 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 olive oil. Crack the eggs into the skillet and fry them over easy over moderate heat, about 3 minutes on one side and about 30 seconds on the other side. Season the eggs with salt and pepper.
On a work surface, spread 8 slices of the whole wheat toast with mayonnaise. Top 4 of these with the lettuce, tomato slices and bacon; top the other 4 with the mushrooms and fried eggs. Stack the sandwiches and cover with the remaining 4 slices of toast. Cut the sandwiches in half and serve.