Heat a wide skillet over medium heat and melt 4 tablespoons of the butter.

Add the sliced onion and season lightly with additional salt and pepper. Reduce to medium-low and cook, stirring regularly, until soft and golden, about 25 to 35 minutes. Lower the heat if the edges scorch, and add a small splash of water if needed to loosen browned bits. Deeper caramelization takes longer. Set the onions aside.

While the onions cook, combine the ground beef, Worcestershire sauce, garlic powder, 1 teaspoon salt, and ½ teaspoon cracked black pepper in a bowl.

Mix gently just until the seasonings are distributed, then divide the beef into 6 equal portions.

On parchment paper, shape each portion into a thin oval slightly wider than a bread slice. Keep the thickness even; the patties will shrink during cooking.

Arrange the 12 bread slices and spread the yellow mustard evenly over 6 of them.

Heat a large skillet or griddle over medium heat. Cook the patties in batches, leaving space between them, for about 3 minutes on the first side, until browned.

Flip and continue cooking, starting to check after about 3 more minutes. Cook each patty to 160°F in the center; insert an instant-read thermometer through the side of a thin patty. Do not press the patties while they cook.

Divide the cooked onions over the patties. Add 1 slice of American cheese and 1 slice of Swiss cheese to each, and allow the cheese to soften briefly.

Transfer each topped patty onto a mustard-coated slice of bread, keeping the onions and cheese facing up.

Remove the pan from the heat briefly and carefully wipe away excess grease with a paper towel held in tongs. Return to medium-low heat for toasting.

Melt 2 tablespoons of the remaining butter in the skillet or across the griddle, dividing it between batches as needed. Add the prepared sandwiches and cover each with a remaining bread slice.

Toast for 2 to 3 minutes, checking the underside, until golden and crisp. Lower the heat if the bread browns before the cheese melts.

Add the final 2 tablespoons of butter beneath the sandwiches as you flip them with a wide spatula, dividing it between batches as needed.

Toast the second side for another 2 to 3 minutes, until golden and the cheese is melted. Press lightly with the spatula to help the sandwich hold together.

Let the patty melts settle for about a minute on a cutting board, then cut each sandwich in half and serve.