Step 1

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.

Step 2

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.)

Step 3

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.

Step 4

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.

Step 5

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