To make the aioli, heat a skillet over medium heat. Add the bacon and cook, turning once, until crisp. Transfer to a paper towel-lined plate and dice when cooled.
Measure out 1 tablespoon bacon fat into a bowl. Add the diced cooked bacon and remaining bacon aioli ingredients and stir to combine well. Taste and adjust seasoning.
Shape the beef into 4 patties. Season with salt and pepper.
Heat the oil in a skillet over medium heat. When hot, add the patties and cook, flipping once, until seared on each side and cooked all the way through, about 8 minutes. Add the cheese after flipping.
Serve the burgers on brioche buns with onion, lettuce, tomatoes, pickles and bacon aioli. Chips on the side.