Melt 2 tablespoons of butter in a cast iron skillet over medium-low heat and add the thinly sliced onions. Cook for 25-30 minutes, stirring occasionally, until deeply golden and soft with browned edges (visual cue: darker amber color).

Season the onions lightly with salt and pepper during caramelizing as they darken. Continue cooking until they look sticky and caramel-browned, not pale.

Divide the ground beef into 4 balls. Season each ball with Worcestershire sauce, garlic powder, salt, and pepper, then press/shape the portions for smashing.

Heat the cast iron skillet over high heat until very hot. Smash each beef ball thin and cook for 2-3 minutes per side, pressing once or twice (visual cue: browned crust on the edges).

Add the remaining butter to the skillet and toast the rye bread slices. Toast over the skillet heat until golden on the outside (visual cue: buttered surface looks crisp and browned).

Stir together mayonnaise, ketchup, Dijon mustard, and Worcestershire sauce in a bowl. Keep mixing until smooth and glossy (visual cue: even pinkish-red color).

Assemble each melt in this order: toasted rye, special sauce, caramelized onions, smashed patty, Swiss cheese, sharp American/cheddar cheese, more onions, and top with toasted rye. Layer the cheeses fully so they cover the patty.

Return the sandwiches to the cast iron skillet and press lightly. Cook 1-2 minutes per side until the cheese is melted and the bread is crisp (visual cue: cheese stretches and the edges sizzle).