1. Heat 1 tablespoon butter in 12-inch nonstick pan over medium heat; add onion. Cook 18 to 20 minutes, stirring until browned. Add a pinch of salt.
2. Toss beef with cheese, mustard, 1 teaspoon salt, pepper, and thyme; shape into four 1/2-inch-thick patties.
3. Heat a grill pan. Grill patties 5 to 6 minutes per side until an instant-read thermometer inserted into side of burger registers 160 degrees F (165 degrees F for turkey).
4. Heat a nonstick skillet. Butter one side of each slice of bread; place buttered side down in skillet, toast. Spread onions on untoasted side of bread; add tomato, burger, and remaining bread; repeat. Makes 4 servings.