For the yogurt-tahini sauce, whisk ingredients in a medium bowl until smooth. Transfer to an airtight container. Cover and refrigerate for up to 5 days.
For the kebabs, whisk oil, garlic and spices in a large mixing bowl. Add chicken. Season with salt and pepper. Mix thoroughly.
Thread chicken onto four metal skewers, dividing evenly. Place in an airtight container. Cover and refrigerate for at least 4 hours or overnight.
Preheat barbecue to medium-high.
Clean and oil barbecue grate. Grill kebabs, turning occasionally until cooked through, 12 to 14 minutes. (If your buns aren’t super fresh, grill them, cut side down, for about 30 seconds.)
Thinly slice radishes and cucumbers.
Smear buns generously with yogurt-tahini sauce and line with radishes and cucumbers. Top with chicken pieces and serve with hot sauce.