In a small mixing bowl, add all of the sauce ingredients. Whisk together until well combined. Set aside.
In a large skillet under medium heat, melt 1 tablespoon of butter and add the onions. Season with salt and sauté, stirring occasionally, until onions are softened and caramelized, about 10-12 minutes. Remove from heat and set aside.
Divide ground beef into 4 equal portions weighing 4 ounces (1/4 pound) and form into a ball. Place each ball of meat onto a piece of parchment paper and flatten into thin, circular patties. Season tops with salt and pepper to taste.
In the same skillet used to make caramelized onions, melt remaining butter over medium-high heat. Add brioche buns or hamburger buns and toast to golden brown. Remove from pan. Set aside until ready to use. Drizzle some oil into skillet and add the patties seasoned side down. Season top of patties with salt and pepper and cook for 2-3 minutes or until patties starts to turn brown with crisp edges. Spread 1 tbsp of mustard on the raw side and then flip. Cook for 1-2 minutes and place a slice of cheese on each of the (4) patties. Cook for an additional minute to allow the cheese to melt. Remove from heat.
Assemble burger by first spreading some of the sauce on both halves of the toasted buns. Top the bottom bun with a tomato slice and 2 lettuce leaves. Stack two patties together and place on top of the lettuce. Spoon a generous amount of caramelized onions over patties and close burger with top half of the bun. Repeat with remaining patties, tomato slice, lettuce and buns.
Serve burgers as is or with regular or animal style fries. Enjoy!