Combine chicken, garlic and lime juice. Let sit for about 15 minutes, then add tandoori paste. Marinate at least 30 minutes; overnight in the refrigerator is even better.
Rub the chicken with the oil. Broil, grill or cook in a nonstick frying pan just until cooked through, about 5 or 6 minutes a side; be careful not to overcook. Transfer to a plate and let cool.
Cut the chicken into thin slices or small bite-size pieces. Set aside.
Mix together mayonnaise and yogurt, then stir in mint, if using. Set aside.
Brush the pitas on one side with water, then heat in a toaster or in a heavy nonstick frying pan. They should puff up, making it easy to split them to form a pocket.
Into each pocket, place a few pieces of chicken, a spoonful of yogurt sauce, a sprinkling of cucumber, a little lettuce, more chicken and so on, until the ingredients are used up or the pitas can't hold any more. Garnish with red onion, tomato and cilantro or mint, then serve.