Combine first 4 ingredients in a bowl; stir well. Add lamb, cheese, and mint; stir well. Divide mixture into 4 equal portions, shaping into 1/2-inch-thick patties.
Prepare grill. Place patties on grill rack coated with cooking spray; grill 5 minutes on each side or until done.
Place patties on bottom halves of buns; top each with 2 tablespoons tomato, 2 tablespoons Cucumber-Dill Sauce, and top half of bun.