Combine seasonings for the marinade (cayenne-olive oil) in a medium bowl. Add the chicken and toss to coat evenly. Marinate for at least two hours or overnight.

Heat 1 teaspoon olive oil in deep saute pan over medium-high heat. Add the chicken and sear on all sides. Remove and set aside.

Add the chicken stock and deglaze the pan, scraping the browned bits on the bottom of the pan. Add the chicken back to the pan along with ½ lemon and ¼ cup olives. Bring to a simmer, cover the pot, and reduce heat to medium-low. Continue to cook until the chicken is tender about 20-25 minutes.

Meanwhile, roast the bell pepper directly on the stove top until the skin has blackened. Transfer to a heat-safe bowl and cover. Let sit for 10 minutes. Carefully remove the skin, discarding the stem and seeds. Chop the pepper.

Remove chicken from heat. Shred the meat, discarding the bones. Toss together with the olives, roasted bell pepper, tomatoes, cilantro. Squeeze the halved lemon and 2 tablespoon of the braising liquid. Season with salt and pepper.

Toast the pita bread. Split open and stuff with the chicken mixture. Serve with hummus if desired.