In a large bowl, whisk together the olive oil, garlic, Meyer lemon juice and zest, oregano, thyme, cumin, paprika, salt, and pepper. Add the chicken and turn to coat every piece. Cover the bowl and refrigerate for at least 1 hour, or preferably overnight, so the flavors really penetrate the meat.

Grate the cucumber on the large holes of a box grater and squeeze out as much liquid as possible using your hands or a clean kitchen towel. In a separate bowl, stir together the grated cucumber, Greek yogurt, garlic, dill, lemon juice, olive oil, and salt and pepper to taste. Cover and chill until you are ready to build the gyros.

Heat a grill or a large cast-iron skillet over medium-high heat. Remove the chicken from the marinade, letting any excess drip off, and cook for 5 to 7 minutes per side, until the exterior is deeply charred and the internal temperature reaches 165°F. Transfer the chicken to a cutting board, let it rest for 5 minutes, then slice it thinly against the grain.

Warm the pita breads on the grill, in a dry skillet, or in a 350°F oven for a couple of minutes until they are soft and bendable. Stack them and wrap in a clean towel or foil to keep them pliable.

Lay one warm pita on a plate and spread a generous layer of tzatziki down the middle. Arrange sliced chicken, red onion, tomatoes, and lettuce on top. Add any optional toppings you like, then fold the pita over the filling or roll it up and serve right away.