Combine Greek yogurt, olive oil, lemon juice, vinegar, garlic, salt, paprika, black pepper, cumin, coriander, oregano, red pepper, and cinnamon in a medium bowl and stir until well combined.
Pound chicken thighs evenly to about 1/2 inch in thickness. Coat all chicken well in marinade, cover, and refrigerate at least 4 hours or overnight.
Preheat the oven to 375 degrees F (190 degrees C). Cut ends off onion and cut onion in half horizontally. Place onion halves next to each other on a parchment or foil-lined rimmed baking sheet. Insert one skewer into each onion half, standing up vertically.
Remove chicken from marinade, allowing excess to drip off, and thread each piece of chicken over the two skewers to form a large stack.
Bake in the preheated oven until evenly browned and cooked through, about 1 hour, rotating halfway through.
Allow meat to rest for 10 minutes before slicing.
Place the sliced chicken in a warm pita topped with tzatziki sauce, lettuce, tomato, and red onion.