Cook the yellow onions in 1 tablespoon butter with 1/4 teaspoon salt over medium-low heat for 35 to 40 minutes until caramel brown.
Shape the beef into two thin patties, season with remaining salt and pepper and cook over medium-high heat for 2 to 3 minutes per side to 160°F.
Layer cheddar, a patty, caramelized onion, drained pickled onion and more cheddar between sourdough slices.
Butter the outside faces with the remaining butter.
Griddle over medium-low heat for 3 to 4 minutes per side until crisp and melted.