In a saucepan, melt 2 tablespoons of the butter with the garlic and thyme on low heat.
While the butter is simmering, start to assemble your sandwiches. Lay your slices of bread out and spread a teaspoon of Dijon mustard onto each slice.
On 2 of the slices of bread, layer the cheese. Add a slice of Gruyère cheese, a slice of Pepper Jack cheese, a slice of cheddar cheese, and a slice of American cheese.
Add the remaining tablespoon of butter to a large skillet and heat over medium heat.
While the skillet is preheating, use a pastry brush to brush the butter from the saucepan onto the top and bottom of both sandwiches.
Gently place your sandwiches into the hot skillet and cook until both sides are golden brown and the cheese is melted. Serve immediately. Enjoy!