1. Cut each mushroom in half crosswise, then cut each in half again, crosswise. Set aside. Cut each zucchini into 1-inch long pieces and set aside. Skewer one piece of chicken onto a bamboo skewer, followed by one piece mushroom, one piece zucchini, chicken, mushroom, zucchini, and finally chicken. Repeat the process with the remaining chicken, vegetables and skewers. Transfer to a plate and refrigerate until ready to use.
2. To make the miso glaze: Combine all the ingredients in a small bowl, stirring until the miso and sugar dissolve. Set aside until ready to use.
3. Preheat a grill to medium-hot coals. Oil the grill to prevent skewers from sticking. Lay the skewers onto the grill, over the burning embers, in a single layer. Baste frequently and turn skewers, until browned all over and chicken is cooked through, 8-10 minutes. Serve warm.
