Preparation

• Combine the tamari, sake, mirin, sugar, honey, and ginger in a 1-quart saucepan; bring to a boil over medium heat, whisking until the sugar dissolves. Remove from the heat and cool completely.
• In a medium bowl, toss the chicken with the marinade. Cover and let sit for 30 minutes.
• Prepare a medium (350°F to 375°F) gas or charcoal grill fire for direct grilling.
• Remove the chicken from the marinade. Return the marinade to the saucepan, bring to a boil, and then turn the heat down to low and simmer until slightly thickened, 5 to 7 minutes. Remove from the heat and set aside.
• Alternately thread 3 pieces of chicken, 3 shiitake, and 3 pieces of scallion onto each skewer, leaving a little space between pieces. Oil the grill grate. Grill the skewers until grill marks form, 2 to 3 minutes. Turn, baste the cooked side with the sauce, and continue to grill, turning and basting, until the chicken is cooked through and the mushrooms are tender, 5 to 6 minutes more. Serve drizzled with the remaining sauce and sprinkled with the sesame seeds and sliced scallions.