Special equipment: 6-inch wooden skewers, soaked in cold water for at least 1 hour
Whisk together the radish puree, mirin, sake, oil, soy and hot sauce in a large bowl. Add the chicken and toss to coat. Cover and refrigerate for at least 2 and up to 8 hours.
Preheat a charcoal grill or kamado-style ceramic charcoal cooker for direct grilling over high heat.
Using 2 skewers for each chicken thigh, thread the edges of the meat onto the skewers. Sprinkle with salt and pepper. Grill until charred and cooked through, about 6 minutes per side. Transfer to a cutting board.
Whisk together the oil, rice vinegar, soy, lime juice and sugar in a large bowl and season with pepper. Pour some over the mizuna, toss to coat and mound in a large shallow serving bowl. Toss the noodles with the remaining dressing and mound in the center of the mizuna.
Cut the chicken thighs in half down the middle between the skewers to make kebabs. Arrange the skewers spoke-like around the mizuna, chicken toward the center, drizzle with additional hot sauce and serve.