In a bowl, combine the yogurt, olive oil, cubed chicken, lemon juice, garlic, shallots, paprika, oregano, chili flakes, and a large pinch each of salt and pepper. Let marinate for 15 minutes at room temperature or up to overnight in the refrigerator.

Preheat the oven to 425° F. Arrange the chicken on a baking sheet. Bake for 15 minutes, toss and bake another 5-10 minutes, or until cooked through. Switch the oven to broil. Broil for 1-2 minutes, until the chicken chars on the edges (watch carefully).

Meanwhile, make the Tzatziki. Combine the feta, ¼ cup yogurt, garlic, and lemon juice in a blender and blend until creamy. Stir in ⅓ cup more yogurt, the cucumbers, and a pinch of salt and red pepper flakes.