In a large cast-iron skillet over medium heat, cook the bacon until crisp, 12 to 14 minutes. Remove to a plate and set aside. Drain the grease from the skillet but do not clean. When cool enough to handle, tear the bacon slices in half.
Meanwhile, mix the mayonnaise and hot sauce to taste in a small bowl. Season with salt and pepper.
Using a biscuit cutter or the rim of a glass, cut a hole in the center of each Texas toast slice.
Melt 2 tablespoons of the butter in the bacon skillet and add 2 of the Texas toast slices. Toast the bread slightly, 1 to 2 minutes. Crack an egg into the hole in each piece of toast and sprinkle the eggs with salt and pepper. Cook the eggs until set on the bottom, 2 to 3 minutes. Using a spatula, carefully flip the toast with the egg, season the other side and spread with the mayonnaise mixture. Top each with a slice of cheese to slightly melt and cook until the egg is the desired doneness. Remove to a cutting board and repeat with the remaining butter, toast, eggs, mayo mixture and cheese.
To build the sandwiches, divide the bacon between 2 of the egg-in-a-hole toast pieces, on top of the cheese. Top each with some tomato slices and lettuce, then top with a remaining egg-in-a-hole toast piece. Serve immediately or wrap in foil to keep warm.