Preheat a waffle iron to medium-high heat. Lightly oil the top and bottom of the waffle iron or coat with nonstick spray.

Melt 2 tablespoons butter in a large skillet over medium high heat. Add onions, garlic and thyme; season with salt and pepper, to taste. Cook, stirring occasionally, until the onions are very soft and caramelized, about 25 minutes; set aside.

Divide ground beef into 4 balls; press each lightly to form 1/4-inch-thick patties; season with salt and pepper, to taste.

Melt remaining 2 tablespoons butter in the skillet. Add patties to the skillet and cook until browned and cooked through to medium rare, about 3-4 minutes per side, flipping only once.

With a butter knife, spread mayonnaise on one side of each slice of bread. Working in batches, place slices, mayonnaise side up, into the waffle iron. Top with patties, onions, cheese and remaining bread slices, mayonnaise side down. Close gently and cook until golden brown and crisp, about 3-4 minutes.

Serve immediately.