In a bowl, combine yogurt, lemon juice, vinegar, olive oil, oregano, paprika, za'atar, cumin, salt and pepper.
Slice chicken breast into thin strips and place in the yogurt mix, stirring to coat.
Allow the chicken to sit for at least 20 minutes, up to overnight
Preheat the oven to 400°
Line a sheet pan with parchment paper. Place sliced onions on one side in a row, sliced bell peppers on the other side, and the marinated chicken in the middle. Try to keep everything in a flat even layer, and use a second sheet pan if it gets too crowded.
Bake on the top rack for 17-20 minutes to get a nice golden brown color on the chicken. Turn on the broiler for 2-3 minutes if desired.
Tzatziki Sauce
Grate the cucumber into a fine mesh strainer. Optional: line the strainer with cheese cloth. Sprinkle salt over the grated cucumber and allow to sit for 10-15 minute to release a lot of the moisture.
Rinse the cucumber to remove the excess salt, and then squeeze in a clothe/cheesecloth, or paper towel to remove as much of the liquid as possible.
In a bowl, add the grated cucumber and the remaining tzatziki sauce ingredients and mix to combine. Season to taste. Feta isn't traditionally in tzatziki, but I add a little to mine for fun!
Chill in the fridge until ready to use.
To assemble the gyros, layer the pita or naan with lettuce, tomatoes, sliced red onion, and a few cucumbers. Top with the warm chicken, peppers and onions. Sprinkle on some crumbled feta and dollop on as much tzatziki as you like.