Caramelize onions
Heat butter and olive oil in a skillet over medium-low. Add onions and cook, stirring occasionally, for 15–20 minutes until soft and deeply golden. Set aside.
Form patties
Shape beef into 4 thin oval patties slightly larger than the bread. Season with salt and pepper.
Cook patties
Cook patties in the same skillet over medium-high for 2–3 minutes per side, until browned and cooked through. Remove and rest.
Make secret sauce
Stir all sauce ingredients together until smooth.
Assemble
Butter one side of each bread slice. On the unbuttered side, layer cheese, patty, onions, secret sauce, more cheese, then top with bread (buttered side out).
Grill
Cook sandwiches in a skillet over medium heat, flipping once, until bread is golden and cheese is melted.
Serve
Slice and serve hot.