In a heavy skillet, melt butter and olive oil over medium heat. Add sliced onions and cook for 5 minutes until softened. Reduce heat to medium-low, add salt, and cook for 30-40 minutes until deeply caramelized and jammy. Set aside.

In a small bowl, whisk together mayonnaise, ketchup, sweet relish, Worcestershire sauce, garlic powder, and black pepper. Refrigerate until ready to use.

Divide ground beef into four equal portions. Shape into oval patties to match the bread shape. Season generously with salt and pepper on both sides.

Heat skillet over medium-high heat. Cook beef patties for 3-4 minutes per side until a brown crust forms and desired doneness is reached. Set patties aside to rest. Wipe out excess grease.

Spread secret sauce on 4 slices of bread. Top each with a slice of Swiss cheese, a cooked patty, a scoop of caramelized onions, and a slice of cheddar cheese. Place the remaining 4 slices of bread on top.

Butter the outside of the sandwiches. Cook in the skillet over medium-low heat for 3-5 minutes per side, gently pressing with a spatula, until the bread is golden and crispy and the cheeses are completely melted.

Remove from skillet, slice in half diagonally, and serve hot.