Heat 2 tablespoons of butter in a large skillet over medium heat.

Cook, stirring occasionally, for about 15-20 minutes, or until the onions are soft and caramelized.

Add the minced garlic and dried thyme, cooking for another 2 minutes until fragrant.

In the same skillet, add the ground beef, breaking it up with a wooden spoon.

Season with salt and pepper and cook over medium-high heat until browned and cooked through, about 7-8 minutes.

Drain any excess fat, then set the ground beef aside.

Spread ½ tablespoon of mayonnaise on one side of each slice of rye bread.

On the non-mayo side of four slices of bread, layer sharp cheddar cheese, followed by a generous amount of the cooked ground beef, caramelized onions, and Swiss cheese.

Top each with another slice of bread, mayo-side out.

Preheat a large skillet or griddle on medium heat.

Melt 1 tablespoon of butter in the pan, then place the sandwiches in the skillet (you may need to do this in batches).

Cook for 3-4 minutes per side, pressing down gently with a spatula, until the bread is golden brown and the cheese is melted.

Add more butter to the skillet as needed when flipping the sandwiches.

Slice the grilled cheese sandwiches in half and serve immediately, while hot and gooey, with a side of pickles or chips for a complete meal.