Melt two tablespoons of butter in a large skillet set over medium-low heat. Once the butter is foaming, stir in the sliced onions. Cook gently, stirring from time to time, until they take on a deep golden hue and soften considerably—this process will take approximately 20 to 25 minutes. Remove from the heat and set aside.

Combine the mayonnaise, ketchup, Dijon mustard, sweet relish, Worcestershire sauce, garlic powder, and smoked paprika (if using) in a small bowl. Mix until the sauce is fully emulsified and smooth. Store covered in the refrigerator until ready to use.

Shape the ground beef into four oval patties that correspond roughly to the dimensions of the bread slices. Season generously with salt and pepper on both sides. In a preheated skillet over medium-high heat, sear the patties for 3 to 4 minutes per side, or until they are cooked through and browned evenly. Remove from the skillet and keep warm.

To assemble the sandwiches, lay out the bread slices with the buttered sides facing down. On the unbuttered interior of four slices, place one slice of cheese, followed by a cooked beef patty. Spoon a portion of the caramelized onions over each patty and spread 1 to 2 tablespoons of the prepared sauce on top. Layer with a second slice of cheese and close the sandwiches with the remaining bread slices, buttered side facing out.

Warm a clean skillet or griddle over medium heat. Grill each sandwich for 2 to 3 minutes per side, pressing gently with a spatula, until the exterior is deeply golden and crisp and the cheese is fully melted.

Transfer to a cutting board, halve diagonally, and serve promptly alongside classic sides such as pickles, fries, or kettle chips.