In a large skillet, cook bacon until crispy over medium heat. Remove to a paper towel lined plate.
To the bowl of a food processor, add mayo, tomatoes, garlic, salt, and pepper. Pulse until tomatoes are chopped and aioli is well combined. Stop and scrape down sides of bowl as needed.
Spread 1/2 tablespoon butter on one side of each of the slices of bread. Toast bread, butter side down, in a skillet until golden.
Spread 1/2 of the aioli on the untoasted side of one slice of bread. Top with lettuce, cheese, and then bacon. Spread remaining aioli on the untoasted side of the other bread slice. Please aioli side down over bacon.