Make the burger sauce: In a medium bowl, combine mayonnaise, ketchup, pickle relish, yellow mustard, white vinegar, garlic powder, and paprika. Mix thoroughly until smooth and well blended. Season with salt and pepper to taste. Set aside.

Caramelize the onions: Heat a cast iron skillet over medium heat. Add 1 tablespoon butter and 1 tablespoon oil. Once melted, add thinly sliced onions and season with salt and pepper. Cook, stirring frequently, until golden brown and caramelized, about 20-25 minutes. Add splashes of water if onions start drying out. Remove to a plate and set aside.

Prepare the patties: Divide ground chuck into 5 equal portions (about 4.8 ounces each). Gently shape into thin patties roughly the same size as your bread slices. Handle the meat as little as possible to keep it tender. Season generously on both sides with salt and pepper.

Cook the patties: Heat a flat-top griddle or cast iron pan over medium-high heat. Add 1 tablespoon oil to prevent sticking. Add burger patties and cook for 3-4 minutes per side for medium-rare, flipping only once. Remove to a plate and let rest.

Assemble the patty melts: Wipe the pan clean and reduce heat to medium-low. Butter one side of each bread slice. Place buttered side down in the hot pan. Layer: cheese slice, beef patty, spoonful of caramelized onions, drizzle of burger sauce, another cheese slice, and top bread slice (buttered side up).

Cook the sandwiches: Cook for 3-4 minutes until the bottom is golden brown and toasted. Carefully flip using a large spatula and cook on the opposite side until golden and cheese is completely melted, another 3-4 minutes.

Serve: Remove from heat and let rest for 1 minute. Cut diagonally and serve immediately with extra burger sauce for dipping if desired.