Melt butter in 12-inch skillet until sizzling; add onions. Cook over medium heat, stirring occasionally, 35-40 minutesuntil onions are tender and deep golden brown.
Stir in thyme, vinegar, 1/4 teaspoon pepper and 1/8 teaspoon salt. Continue cooking, stirring occasionally, 5 minutes or until onions are caramelized.
Heat gas grill on medium or charcoal grill until coals are ash white.
Combine ground beef, mustard and salt in bowl; mix lightly. Shape into 4 (3/4-inch thick) patties. Sprinkle both sides of patties with pepper.
Place patties onto grill. Grill, turning once, 10-12 minutes or until internal temperature reaches at least 160°F or until desired doneness.
Grill hamburger buns until toasted. Fill buns with hamburgers; top each with 1/4 cup onion mixture.
*Substitute 1/4 teaspoon dried thyme leaves.
Broiling Directions: Prepare onions and patties as directed above. Heat broiler. Place patties onto broiler pan. Broil, turning once, 12-15 minutes or until internal temperature reaches at least 160°F or until desired doneness.