Make the chicken marinade: in a 9x13-inch baking dish, whisk together 1/2 cup Greek yogurt, 1/4 cup olive oil, 1/4 cup lemon juice, 5 minced garlic cloves, 1 tablespoon smoked paprika, 1 tablespoon chopped oregano and a pinch each of Kosher salt and black pepper.
Add the chicken breasts to the dish and toss to coat in the marinade. Cover and refrigerate for at least 1 hour or up to 12 hours.
Preheat the oven to 425°F (220°C). Remove marinated chicken from the fridge and let sit at room temperature while the oven preheats.
Roast the chicken in the baking dish for 20 to 25 minutes, until cooked through. Remove from oven and let rest 5 minutes, then shred with two forks or thinly slice.
While the chicken roasts, prepare the fries. In a small bowl whisk together 1/4 cup olive oil, 2 minced garlic cloves, 3 tablespoons chopped oregano, 1/2–1 teaspoon cayenne, and salt and pepper to taste.
Place the potato matchsticks in a large bowl and pour half of the garlic-oregano mixture over them; toss gently to coat. Spread the potatoes in a single layer across two baking sheets.
Bake the fries at 425°F for 15 to 20 minutes, then flip the fries, reduce oven temperature to 400°F, and bake another 15 to 20 minutes until golden and crisp.
When fries are done, toss them with the remaining garlic-oregano mixture and 1/2 cup crumbled feta.
Warm the pitas or naan. Stuff each with shredded chicken and desired tzatziki, pickled red onions, radishes, and lettuce. Serve with the feta fries on the side.