How to Make It
Bring vinegar, 1/2 cup water, and sugar to a boil in a small saucepan, stirring to dissolve sugar. Place bell peppers, cucumber, onion, and 2 tablespoons dill in a large bowl; pour hot vinegar mixture over vegetables. Let stand 30 minutes, stirring occasionally. Drain and discard liquid.
Combine remaining 1 tablespoon dill, beef, salt, and pepper. Shape beef mixture into 4 (1/2-inch-thick) patties. Heat a grill pan over medium-high heat. Coat pan with cooking spray. Add patties to pan; cook 3 minutes. Turn patties over; cook 2 1/2 minutes. Top patties with cheese; cook 1 1/2 minutes or until desired degree of doneness.
Spread mayonnaise over bottom halves of buns. Top with patties. Divide vegetable mixture over patties; top with top halves of buns.
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