Select twelve 10-to 12-inch wooden or metal skewers; if using wooden skewers, soak them in water for 30 minutes and drain before using.
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.
at 9:38 PM on Jun. 29, 2009