Make the tzatziki first: grate and squeeze the cucumber dry, then stir into the yogurt with garlic, olive oil, lemon juice, dill, and salt. Refrigerate.
Combine lamb, grated onion, garlic, oregano, mint, cumin, salt, pepper, and feta. Mix by hand just until combined, no more than 30 seconds.
Divide into 4 patties, pressing a shallow dimple into the center of each.
Let the patties rest at room temperature for 10 minutes before cooking.
Cook in a hot cast-iron pan or on a grill, 4 to 5 minutes per side, without pressing down.
Rest the burgers for 3 minutes off the heat.
Warm the pita or buns and assemble with lettuce, burger, tzatziki, tomato, and red onion. Serve with lemon wedges.