For the tzatziki Toss cucumber with salt in colander and let drain for 15 minutes. Whisk yogurt, oil, mint, and garlic together in bowl, then stir in cucumber. Cover and refrigerate for at least 30 minutes to allow flavors to meld. Season with salt and pepper to taste.
For the wraps Combine lemon juice, oregano, garlic, salt, and pepper in large bowl. Add lamb and gently mix with your hands until thoroughly combined. Pinch off and roll mixture into 12 balls. Gently flatten balls into round disks, about ½ inch thick and 3 inches in diameter.
Heat oil in 12-inch nonstick skillet over medium-high heat until just smoking. Add patties and cook until lightly browned, 6 to 8 minutes, flipping halfway through cooking. Transfer patties to paper towel– lined plate.
Using spoon, spread tzatziki evenly inside each lettuce leaf. Divide patties among leaves and top with tomato pieces and feta. Serve.