Place a large skillet or sauté pan over medium heat. Add 4 tablespoons of butter and allow it to melt.

Add the sliced onion and season generously with salt and pepper. Reduce the heat to medium-low and cook for 15 to 20 minutes, stirring regularly, until the onions are soft, golden, and caramelized. Remove the skillet from the heat.

While the onions cook, add the ground beef to a large bowl. Add the Worcestershire sauce, garlic powder, 1 teaspoon salt, and ½ teaspoon cracked black pepper.

Gently mix the beef with your hands until the seasonings are evenly distributed. Press the meat evenly into the bottom of the bowl and divide it into 6 equal portions.

Lay a large sheet of parchment paper or foil on your work surface. Shape each portion into a very thin oval patty that is slightly larger than a slice of bread, since the meat will shrink during cooking.

Arrange the 12 slices of bread on your work surface. Spread the yellow mustard evenly over 6 slices.

Heat a griddle or large skillet over medium heat. Arrange the patties on the hot surface, working in batches if necessary, and cook for about 3 minutes on the first side.

Flip each patty. Divide the caramelized onions evenly among the patties, spreading them across the beef.

Place 1 slice of American cheese and 1 slice of Swiss cheese over each onion-topped patty. Continue cooking until the second side of the beef has cooked for about 3 minutes and the cheese begins to melt.

Carefully transfer each beef patty, onion side facing up, onto one of the mustard-coated bread slices.

Carefully wipe excess grease from the griddle using a paper towel held with a spatula or tongs.

Melt 2 tablespoons of butter across the griddle. Place the prepared patty melts onto the buttered surface and top each one with a remaining slice of bread.

Toast for 2 to 3 minutes, watching closely, until the bottoms are crisp and golden brown. Reduce the heat if the bread is browning too quickly.

Add the remaining 2 tablespoons of butter to the griddle. Flip each sandwich as the butter melts underneath it.

Toast the second side for another 2 to 3 minutes, or until golden and crisp and the cheese is fully melted.

Transfer the patty melts to a cutting board. Let them rest briefly, slice each sandwich in half, and serve immediately.