How to Make It
In a baking dish, toss the chicken with the olive oil, garlic, paprika and Aleppo pepper until well coated. Cover and refrigerate for at least 2 hours or overnight.
In a medium bowl, whisk the yogurt with the crème fraîche, lemon juice, olive oil, garlic and vinegar. Fold in the cucumber and radishes and season with salt and pepper. Cover and refrigerate the tzatziki until well chilled, about 1 hour.
In a medium saucepan, cover the potatoes with water and bring to a boil. Add a generous pinch of salt and simmer over moderate heat until the potatoes are tender but not falling apart, about 15 minutes. Drain and transfer to a baking sheet to cool completely. Using your hand, gently smash the potatoes into 1/2-inch-thick rounds.
In a large, deep skillet, heat 1/4 inch of vegetable oil until shimmering. Add the potatoes and fry over moderately high heat, turning once, until golden and crisp, 7 to 10 minutes. Transfer to paper towels to drain. Season with salt and pepper and keep warm.
Preheat the oven to 400°. Preheat a cast-iron griddle. Thread the chicken skin side up onto 12 skewers (see Note) and season with salt and black pepper. Arrange the chicken skewers skin side down on the griddle and cook over moderately high heat until the skin is crisp, about 5 minutes. Transfer the skewers to a baking sheet and roast in the oven for 10 to 12 minutes, until the chicken is cooked through. Serve the kebabs with the crispy potatoes and tzatziki.
Aleppo pepper is a moderately hot crushed dried chile from Turkey and Syria; it is available at gourmet markets and penzeys.com. If using bamboo skewers, soak them in water for at least 30 minutes before threading.