In a large bowl, whisk together Greek yogurt, lemon juice, olive oil, red wine vinegar, minced garlic, oregano, cumin, paprika, coriander, salt, pepper, and red pepper flakes until smooth.
Add chicken thighs to the marinade, turning to coat thoroughly. Place in a ziplock bag and massage marinade into meat. Refrigerate for at least 2 hours, ideally 4-12 hours.
Preheat grill or cast-iron skillet over medium-high heat. Oil grates if using grill. Remove chicken from marinade, letting excess drip off.
Cook chicken for 5-7 minutes per side for thighs, or 4-6 minutes per side for breasts, until internal temperature reaches 165°F and edges are nicely charred.
Transfer cooked chicken to a plate and let rest for 10 minutes. Slice thinly against the grain.
Warm pita breads in a dry skillet for 20-30 seconds per side until pliable.
Assemble gyros: Spread tzatziki on warm pita, add sliced chicken, lettuce, tomatoes, red onion, cucumber, and feta cheese. Fold and serve immediately.