Stir the yogurt, cucumber, half the feta, lemon juice and garlic together and chill the sauce.
Mix the lamb, oregano, salt, pepper and remaining feta gently, then shape 4 thick patties with shallow center dimples.
Heat a cast-iron skillet over medium-high heat and sear the patties for 4 minutes on the first side.
Flip, lower the heat to medium and cook for another 4 to 6 minutes without pressing until the centers reach 160°F.
Rest the patties for 3 minutes while the buns warm cut-side down in the skillet.
Spread cucumber feta sauce on each bun and add romaine, tomato and one lamb patty.