Heat olive oil in a 10-inch nonstick Dutch oven over medium heat. Add onions, carrots and bell peppers and saute until soft, about 2 minutes. Add garlic and stir until well mixed. Add cayenne, cumin, paprika, chili powder, oregano, and white pepper; saute for 1 minute. Add the turkey legs and stir at least 30 seconds so that legs are completely coated in the spices. Cover with chicken broth and bring to a simmer. Reduce heat to low and cover. Cook for 4 hours or until meat is tender and separates easily from the bone. Shred the turkey drumsticks and toss the meat with the dipping sauce. Stuff the pitas with the turkey mixture. Serve immediately.
In a medium bowl, mix all ingredients together. Cover and refrigerate for at least 1 hour so that flavors meld. Bring to room temperature before tossing the turkey in the sauce.