In a large skillet over medium-low heat, melt 2 tablespoons of butter. Add the sliced onions and cook slowly, stirring occasionally, for 25-30 minutes until deeply golden brown and sweet. Remove from skillet and set aside.

In a medium bowl, gently combine the ground beef, Worcestershire sauce, salt, and pepper. Do not overmix. Form the mixture into four thin, oval-shaped patties slightly larger than your bread slices.

Wipe out the skillet and place it over medium-high heat. Cook the patties for 3-4 minutes per side, or until browned and cooked through. Transfer to a plate.

Wipe the skillet clean. Butter one side of each slice of rye bread. For each sandwich, place one slice butter-side down in the skillet over medium heat. Top with one slice of cheese, a beef patty, caramelized onions, and another slice of cheese. Place the second slice of bread on top, butter-side up.

Grill for 3-4 minutes per side, until the bread is golden and crispy and the cheese is fully melted. Serve immediately.