Heat a cast iron skillet over medium high heat. Add the bacon and cook until the fat renders and the bacon is crispy, 8-10 minutes. Remove the bacon from the skillet using tongs and set onto a paper towel lined plate to drain.
Toast the bread, then spread half a tablespoon of aioli onto each slice of bread.
Top two slices of bread with a quarter cup of sprouts each, then two slices of butter lettuce each, then 2 slices of bacon each, then 2 fried green tomatoes.
Drain all but 1 tbsp bacon fat from the skillet and reheat over medium heat. Gently crack two eggs into the skillet and fry until the whites are set but the yolks are still runny, 4-5 minutes.
Remove the fried eggs with a spatula and place one egg onto each sandwich. Top with the sandwiches with the remaining two pieces of bread. Slice in half, and serve immediately.