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.