Season the chicken in a bowl with the olive oil, greek seasoning, and kosher salt. Heat a cast iron skillet over medium flame and sear the chicken for 3 minutes on each side. Temperature should read 165° when done.
Peel, seed, and grate the cucumber with a cheese grater. Use a mesh cloth or strainer to press as liquid from the cucumber shreds as possible. Add the shreds to a bowl with the yogurt, olive oil, garlic, dill, cumin, salt, and white pepper. Mix well and refrigerate until needed.
To serve build individual chicken gyro according to taste using the pita, tzatziki, and garnishes. See post for suggestions.