Combine 1 cup yogurt, turmeric, paprika, and oregano in a bowl. Sprinkle chicken pieces with salt and pepper, add to a large zip-top bag or sealed container with yogurt marinade, toss to coat. Refrigerate at least 1 hour or overnight.
To make the skewers, remove chicken from the fridge for 10 minutes. Wipe off excess marinade. Thread onto skewers, adding lemon slices between every few chicken pieces. Cook on a grill pan or grill heated to medium heat, rotating every 5 minutes, until chicken reaches 160 degrees. Allow to sit for five minutes before serving. Top with a sprinkle of salt.
Meanwhile, make the yogurt sauce: Mix lemon juice, remaining 1/3 cup yogurt, chili oil and a pinch of salt in a small bowl. Add 1 tsp of water to thin sauce, if needed.
Serve chicken on greens or on a platter, drizzle with yogurt sauce, as desired.