Caramelize onions in 2 tbsp butter over medium heat for 15–20 minutes. Set aside.
Shape beef into 4 patties, season, and sear 3–4 minutes per side. Set aside.
Mix all sauce ingredients in a small bowl and stir until smooth.
Lay out 4 slices of rye bread and spread sauce on each. Add cheese, patty, onions, more cheese, and top bread. Butter outsides.
Toast in skillet until golden and cheese is melted. Slice and serve with extra sauce.
