In a large skillet over medium heat, add the olive oil and sauté the chopped onion until translucent, about 5 minutes. Add the minced garlic and cook for an additional minute until fragrant.
Add the ground beef or lamb to the skillet, breaking it apart with a spatula. Cook until browned, about 5-7 minutes. Drain any excess fat if necessary.
Stir in the cumin, coriander, paprika, cinnamon, salt, black pepper, and cayenne pepper (if using). Cook for another 2-3 minutes until the spices are well combined and fragrant. Remove from heat and stir in the chopped parsley. Allow the mixture to cool slightly.
Preheat your grill or grill pan over medium-high heat. While it heats, take each pita round and cut a pocket into the side.
Stuff each pita with the meat mixture, pressing gently to ensure the filling is packed in.
Place the stuffed pitas on the grill and cook for about 3-4 minutes on each side, or until the pitas are golden and slightly crispy.
Remove from the grill and let cool for a minute. Serve warm with plain yogurt and lemon wedges on the side.