To make the glaze: Combine the vegetable stock, sake, soy sauce, Mirin, brown sugar, rice vinegar, garlic, and ginger in a medium non-reactive saucepan and bring to a boil over high heat. Dissolve the cornstarch in the water and whisk into the soy mixture. Boil for 1 minute. Remove from heat and set aside.
Preheat a grill to medium high or the oven to 450 degrees F.
To assemble: Thread 5 mushrooms, lengthwise (starting with the bottom of each stem), onto each of 4 skewers. Thread 5 pieces of squash onto each of 4 other skewers and thread 7 red pepper pieces onto each of 4 other skewers. Place the kabobs on a baking sheet and brush with the vegetable oil, rotating the kabobs to coat evenly. Place the kabobs on the grill, grouping each type of vegetable for consistent cooking times. Close the grill and cook until the vegetables are tender, 10 to 20 minutes. (The vegetables have slightly different cooking times, so check th
at 8:52 PM on Aug. 26, 2010