Prepare onion rings according to package directions; keep warm on a wire rack in a jelly-roll pan at 200° up to 20 minutes.
Stir together tomatoes, honey, cider vinegar, 1/4 cup onion, and 1/4 tsp. salt.
Preheat grill to 350° to 400° (medium-high) heat. Gently combine beef, pepper, and remaining 1/4 cup onion and 1/2 tsp. salt in a large bowl until blended, using hands. Shape mixture into 4 (4-inch-wide, 3/4-inch-thick) patties.
Grill, covered with grill lid, over 350° to 400° (medium-high) heat 5 to 6 minutes on each side or until beef is no longer pink in center. Grill ham slices 1 to 2 minutes on each side or until lightly crisp. Grill buns, cut sides down, 1 to 2 minutes or until lightly toasted.
Layer each of 4 bun halves with lettuce, burger, Chowchow or Banana Pepper Spread, grilled ham slices, and an onion ring. Spoon tomato mixture into centers of each onion ring. Top with remaining bun halves. Serve with remaining onion rings.
Note: For testing purposes only, we used Alexia Onion Rings with panko coating.