PreparationCaramelize Onions: Heat 2 tablespoons vegetable oil in a medium skillet over medium heat and cook 2 thinly sliced large onions, stirring often and adding water as needed to prevent burning, until deep golden brown and very soft, 20-25 minutes. Make & Cook Patties: Gently mix 1/4 small onion, finely chopped, 1 pound ground beef chuck (20% fat), 1 tablespoon ketchup, 1/2 teaspoon garlic powder, 1/4 teaspoon freshly ground black pepper, and 1 teaspoon kosher salt in a medium bowl. Divide into 4 portions and press each between 2 pieces of parchment or waxed paper until about 1/4 " thick (you want them roughly the same dimensions as the bread you're using). Heat 1 tablespoon vegetable oil in a large skillet, preferably cast iron, over medium-high. Working in 2 batches, cook patties, pressing gently, until browned but still pink in the center, about 2 minutes per side. Transfer to a plate. Do ahead: Patties can be formed 8 hours ahead. Cover and chill. Assemble & Press Melts Wipe out skillet; reduce heat to medium. Top 4 slices seeded rye bread with 4 ounces thinly sliced aged sharp cheddar, then patties, onions, 4 ounces thinly sliced Swiss cheese, and another 4 slices bread; spread tops with 1 teaspoon mayonnaise each. Cook, mayo side down and weighted with a foil-covered heavy pan, until golden brown, about 3 minutes. Remove pan; spread top of each melt with 1 teaspoon mayonnaise. Flip; weight with pan. Cook until bread is golden brown and cheese is melted, about 3 minutes.