Mix Greek yogurt, olive oil, lemon juice, garlic, shallots, smoked paprika, oregano, pepper flakes, salt, and pepper.
Coat the chicken cubes and marinate for 15 minutes or up to overnight.
Preheat oven to 425°F.
Spread sliced peppers on a sheet pan with olive oil, salt, and pepper; roast 18–20 minutes.
Place chicken on another sheet pan and bake for 15 minutes; stir and cook 5–10 minutes more.
Broil chicken for 1–2 minutes to lightly char.
Blend feta, 1/4 cup yogurt, garlic, and lemon juice until smooth.
Stir in remaining yogurt, cucumber, salt, and optional pepper flakes.
Warm pitas and spread tzatziki inside.
Add lettuce, chicken, peppers, pickled onions, herbs, and extra tzatziki.