Preheat the oven to 400 degrees F. Line a baking sheet with parchment paper. Lay the bacon side by side on the baking sheet and transfer to the oven. Bake for 15-20 minutes, until the bacon is browned and crispy. Transfer to a bed of paper towels to drain.
Slice an heirloom tomato; transfer slices to a bed of paper towels and sprinkle with salt. Let stand for 10 minutes.
To a non-stick skillet, melt the butter. Add the slices of bread (you may have to do these one at a time if your skillet isn't big enough), and toast on both sides for 1-2 minutes. Set aside.
Wipe the skillet clean with a paper towel and add a few tablespoons of olive oil. Place over medium-high heat. When the oil is hot add the egg. Fry on one side for 1-2 minutes. Sprinkle with salt and pepper. Repeat with the second egg.
To assemble sandwiches, add a few swipes of mayonnaise to each piece of bread. Add the sprouts to the bottom halves of bread, top with tomato slices, bacon and then fried eggs. Place the reserved halves of bread on top of the sandwich and press down slightly so the yolk breaks. Serve immediately.