In a small bowl, combine the softened butter and mayonnaise until smooth.

In a medium bowl, combine the ground beef, onion powder, garlic powder, 1 teaspoon of the salt, pepper, and Worcestershire. Divide the beef mixture into 4 equal portions. Using your hands, form each portion into 4 thin patties roughly the shape of the bread. Sprinkle the patties with the remaining ½ teaspoon salt.

Heat a skillet over medium heat. Working in batches, add the patties and cook until no longer pink in the middle, 3-4 minutes per side. Transfer to a plate.

Spread one side of each sourdough slice with 1 tablespoon of the mayonnaise mixture.

Wipe out the skillet and return it over medium-low heat. Top half the sourdough slices, dry sides up, with 1 cheese slice each, 1 cooked patty each, then top evenly with the caramelized onions. Top each sandwich with the remaining slices of cheese, and the remaining sourdough, dry sides down.

Add the sandwiches to the skillet and cook until the bread is golden and the cheese is melted, 4-5 minutes per side. Cut each patty melt in half and serve immediately.