Preheat a grill or grill pan on medium heat. Soak the wooden skewers in water.
Place all marinade ingredients in a bowl and mix, let sit 20 minutes.
Drizzle the pita with 1/3 of the olive oil. Season with salt, pepper, and lemon zest.
In a blender add the roasted peppers, broth, lemon juice, and remaining olive oil. Blend until smooth. Season with salt and pepper to taste.
Skewer the chicken onto the wooden skewers. Grill until all sides have nice grill marks and fully cooked to an internal temperature of 160 degrees using a thermometer, about 8 - 10 minutes. (Alternatively, you can bake the chicken skewers in a 375 degree F oven)
Grill the pita for 1 - 2 minutes per side. Cut into quarters.
Drizzle the red pepper sauce over top and serve with pita.