Grind flank steak. Add water, room-temperature butter and 1/2 chopped red onion and garlic. Season with salt and pepper. Form burgers and make an incision in the middle of each patty. Gently rub with oil. Fry on a degreased grill pan for 3 minutes on each side.
Slice a bun horizontally and fry on a pan until golden brown. Spread ketchup over one half of the bun and mayonnaise over the other half. Slice tomato, onion, and cucumber. Place meat on one half of the bun, add cheese, sliced vegetables, and lettuce. Assemble both halves and serve.