Prepare the naan bread according to the recipe instructions. Keep the bread covered.
Marinate the chicken.
In a large bowl or zip-top plastic bag, place all the ingredients for the chicken except for the chicken itself and mix to combine well either with a whisk, or by closing the plastic bag securely and massaging the mixture from outside the bag.
Place the chicken breasts in the marinade, and either cover the bowl tightly or close the bag securely. Place in the refrigerator and marinate for 1 hour or for up to 2 days.
Make the tzatziki.
Place the shredded cucumber in a clean tea towel (a kitchen towel with a flat, non-linty weave), enclosed the cucumber tightly and squeeze tightly to remove as much moisture as possible from the cucumber.
Place the cucumber in a medium-size bowl, add the remaining tzatziki ingredients, and whisk to combine well. Sprinkle the top with a bit more chopped fresh dill, cover the bowl tightly and refrigerate until ready to serve.
Cook the chicken.
Remove the chicken breasts from the marinade, and place side by side on a rimmed baking sheet lined with aluminum foil and greased with cooking spray. Discard any remaining marinade.
Place the chicken directly under your oven’s broiler and cook for 6 minutes. Remove the chicken from the oven and flip each breast over carefully.
Return to the oven and broil until cooked through (4 to 6 minutes more, depending upon the thickness of the chicken breasts). Remove the chicken from the oven and allow to rest for at least 5 minutes before slicing each breast into strips.
To make each individual gyro, pile strips of chicken, tzatziki sauce, tomatoes, sliced red onion and crumbled feta cheese in the center of a warm piece of gluten free naan. Fold the bread in half and serve.