Prepare grill or broiler.
To prepare onions, heat olive oil in a large nonstick skillet over medium-high heat. Add onion and 1/4 teaspoon salt; sauté 12 minutes or until golden brown. Stir in sugar and vinegar; cook 30 seconds. Remove from pan.
To prepare burgers, heat pan coated with cooking spray over medium-high heat. Add bell pepper and celery; sauté 3 minutes or until tender.
Combine the bell pepper mixture, crackers, 1/4 cup ketchup, thyme, 1/4 teaspoon salt, beef, and egg in a large bowl. Divide mixture into 6 equal portions, shaping each into a 1/2-inch-thick patty.
Place patties on a grill rack or broiler pan coated with cooking spray; cook 5 minutes. Carefully turn patties over; brush with remaining ketchup. Cook 5 minutes or until done.
Place 1 patty on each of 6 bread slices. Top each patty with 1/4 cup onion mixture and 1 bread slice.