Marinate the chicken: In a large bowl, add the chicken, yogurt, olive oil, zest and juice of the lemon, garlic, dill, salt, pepper, paprika, and oregano. Stir until the chicken is well coated. Cover and marinate for 20 minutes.
Make the sauce: In a medium bowl, stir together the yogurt, cucumber, dill, parsley, garlic, lemon juice, and salt until combined. Refrigerate until ready to serve.
Preheat your grill to medium-high heat, about 400 degrees F. Clean the grill grates.
Prepare the vegetables: Drizzle the vegetables with olive oil and season with salt and pepper. Toss to coat.
Cook the chicken and vegetables: Grill the chicken and vegetables for 6 to 8 minutes per side, until the chicken reaches 165 degrees F. Remove and let rest.
Warm the pita on the grill for 2 minutes per side.
Chop the chicken and vegetables into bite-sized pieces.
Serve: In a bowl, add yogurt sauce, grilled chicken, and vegetables, topped with pickled red onion and fresh herbs.