Combine all aioli ingredients in medium bowl and mix well. Set aside.
Melt butter in large saute pan over medium-low. Add onions and stir occasionally (adding more butter if onions start to brown too deeply) until onions are a nice golden-brown, about 25 minutes.
Meanwhile, cook bacon in saute pan or cast-iron pan over medium-high for 2-3 minutes per side or until bacon has reached desired crispiness.
Separate ground chuck into four patties of equal size, about 5 inches in diameter and 1 inch tall. Try not to handle the burgers too much, as this can make them tough. Season one side of patties with half the salt and pepper.
Prepare your grill. If using a charcoal grill, bank coals to one side for high heat. On a gas grill, cook the burgers over high heat. Alternatively, heat 1 tablespoon olive oil or leftover bacon grease in skillet over medium-high.
Place your burgers on the grill or pan, seasoned side down. Season the other side of the burgers with remaining salt and pepper.
If cooking on grill, cook burgers 2-4 minutes per side, depending on desired doneness. On a stovetop, cook 3-5 minutes minutes per side, adjusting as needed. After flipping, top patties with cheese for maximum meltiness. Once cooked, place burgers on plate to cool.
Once cooled, place each patty on a lettuce leaf. Each patty will likely need about two lettuce leaves to hold it together.
Place bacon and onions on patties and top with generous dollop of aioli. Wrap lettuce around burger and serve!