Caramelize the Onions:

In a skillet over medium heat, melt 1 tablespoon of butter. Add the sliced onions and a pinch of salt. Cook, stirring occasionally, for about 20-25 minutes, or until the onions are soft, golden, and caramelized. Reduce heat if onions start to brown too quickly. Once caramelized, remove from the skillet and set aside.

Form and Cook the Patties:

While the onions are cooking, form the ground beef into two equal-sized patties, slightly larger than your bread slices to account for shrinkage during cooking. Season each side with salt and pepper.

In the same skillet, heat 1 tablespoon of vegetable oil over medium-high heat. Cook the patties for about 4-5 minutes per side, or until they reach your desired level of doneness. Remove from the skillet and set aside.

Assemble the Patty Melts:

Butter one side of each slice of bread. Place one slice of cheese on the unbuttered side of two slices of bread.

Place a cooked beef patty on top of the cheese, followed by half of the caramelized onions. Top with another slice of cheese and the remaining slices of bread, buttered side up.

Grill the Patty Melts:

Heat the skillet over medium heat. Place the assembled sandwiches in the skillet. Cook for about 3-4 minutes on each side, or until the bread is golden brown and crispy, and the cheese is fully melted.

Serve:

Once both sides are golden and the cheese is melted, remove the sandwiches from the skillet. Let them cool slightly before slicing in half.