In a food processor, pulse together chicken, eggs, garlic, 2 tablespoons ginger, scallions, salt, flour, cornstarch and panko until well combined. Form chicken mixture into 1 1/2-inch balls. To prevent mixture from sticking to your fingers, wet hands with water between rolling balls. Place balls on a plate, without touching.
Meanwhile, roughly chop the 2-inch piece of ginger and in a medium pot over high heat, bring water to boil and add ginger. Place 8 to 10 balls at a time into the water. Cook for about 6 minutes. (Balls will float to top and change color, but inspect for pink on outside.) Remove from water using a slotted spoon and drain on paper towels. Repeat until all balls are cooked. When cool enough to handle, thread 3 balls per bamboo skewer and set aside.
To make the yakitori sauce, whisk together sake, soy sauce, mirin, honey and cornstarch in a small saucepan. Bring to a boil over medium-high heat, reduce heat to medium and simmer, stirring occasionally, until thickened, about 2 minutes.
Preheat gas grill to medium heat. Place skewers on lightly oiled grill. (Wipe oil on with a paper towel.) Cook 7 to 10 minutes, turning gently as they brown lightly. When chicken is done, lightly baste balls with the yakitori sauce. Place skewers on serving plate, drizzle with additional yakitori sauce. Sprinkle with cilantro, green onion, sesame seeds and cayenne pepper. Serve immediately with Sweet Chili Dipping Sauce.