Prepare tzatziki sauce: In a bowl, mix Greek yogurt, grated cucumber, minced garlic, olive oil, lemon juice, fresh dill, salt, and pepper. Chill until ready to serve.
Marinate beef: In a bowl, toss sliced beef with olive oil, paprika, garlic powder, oregano, cumin, salt, black pepper, and lemon juice. Let sit 10 minutes.
Cook beef: Heat a skillet over medium-high heat. Sear beef in batches 2–3 minutes per side until browned and tender. Remove from heat.
Assemble gyros: Warm pita breads. Layer with shredded lettuce, sliced tomatoes, onions, cooked beef, tzatziki sauce, and optional feta.
Serve: Fold gyros, serve with lemon wedges on the side. Enjoy immediately.