In a medium bowl, combine Greek yogurt, lemon juice, grated cucumber, chopped dill, and salt and pepper. Stir well and refrigerate for at least 15 minutes.

Heat olive oil in a skillet over medium heat. Add minced garlic and sauté until fragrant, about 30 seconds.

Add ground beef and break apart with a wooden spoon. Cook until browned, about 5 to 7 minutes.

Stir in oregano, cumin, paprika, salt, and black pepper. If desired, add cayenne pepper. Cook for an additional 2 to 3 minutes.

Warm pita bread in the skillet for about 1 minute on each side or wrap in foil and warm in the oven.

Assemble gyros by placing beef mixture in warm pita, topping with onion, tomato, lettuce, and a generous amount of tzatziki.

If desired, sprinkle feta cheese on top before serving.