In a mixing bowl, combine Greek yogurt with fresh lemon juice, grated cucumber, chopped dill, and a pinch of salt and pepper. Stir until well blended and refrigerate for 15 minutes.

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

Add the ground beef, breaking it apart with a spatula. Cook for 5-7 minutes until browned and fully cooked.

Once browned, sprinkle in oregano, cumin, paprika, salt, black pepper, and optional cayenne. Stir to combine and cook for an additional 2-3 minutes.

Warm pita bread in the skillet for about 1-2 minutes on each side or wrap in foil and heat in a preheated oven at 350°F for 5 minutes.

Assemble each gyro by placing a scoop of spiced beef in the center of each warm pita, layering in onion, tomato, and lettuce, then topping with tzatziki.

Sprinkle feta cheese over the assembled ingredients, roll up the pita to secure filling, and serve immediately.