Add sauce ingredients in a bowl and mix to combine. Set aside.

Heat a cast iron skillet over medium heat. Add 1 Tablespoon butter and 1 Tablespoon oil and once melted, add onions then season with salt and pepper. Cook, stirring frequently, until golden brown and caramelized, about 20 - 25 minutes. Add a splash of water as needed, if the onions seem to be drying out. Remove to plate and set aside.

Divide meat into five patties (4.8 oz each) and shape each into thin patties, roughly the same shape as the bread slices. Handle the meat as little as possible. Season patties generously on both sides with salt and pepper.

Heat a flat-top griddle or cast iron pan over medium-high heat. Grease with 1 tablespoon oil. Add burger patties and cook for 3-4 minutes, flipping once, until medium-rare. Remove to plate to rest.

Wipe griddle/pan clean. Reduce pan heat to medium-low. Butter one side of each piece of bread. Place buttered side of bread face down in hot pan. Top with a slice of cheese, beef patty, a spoonful of onions, drizzle of burger sauce, another slice of cheese, and top slice of bread (butter side facing up). Cook for a few minutes, until bread is toasted on the bottom. Gently flip and cook on opposite side until golden, and cheese is melted.

Serve with leftover burger sauce for dipping, if desired.