Heat oil in a medium skillet over medium heat and cook onions, stirring often and adding water as needed to prevent burning, until deep golden brown and very soft, 20–25 minutes. Set aside.

Gently mix onion, beef, ketchup, garlic powder, salt, and pepper, in a medium bowl. Divide into 4 portions and press each between 2 pieces of parchment or waxed paper until about ¼” thick (you want them roughly the same dimensions as the bread you're using.)

Heat oil in a large skillet, preferably cast iron, over medium-high heat. 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.

Wipe out skillet and reduce heat to medium. Top 4 slices of bread with cheddar, then beef patties, caramelized onions, and Swiss cheese. Close up sandwiches and spread each top with 1 tsp. mayonnaise. Place in pan, mayonnaise side down, and weight with a foil-covered heavy pan. Cook until bottom slice is golden brown, about 3 minutes. Remove weighted pan and spread the top of each sandwich with 1 tsp. mayonnaise. Flip and weight again. Cook until other side is golden brown and cheese is melted, about 3 minutes.

DO AHEAD: Patties can be formed 8 hours ahead. Cover and chill.