Kinoko Mochi

Name: Kinoko Mochi
Ingredients: 2 tablespoons vegetable oil; 1 cup broccoli florettes; 1 cup summer squash, halved and sliced; ½ cup red bell pepper julienned; 5 shishito peppers, halved lengthwise; ¼ Japanese eggplant, sliced; 2 shiitake mushrooms, julienned; 1 royal trumpet mushrooms, sliced; ¼ bunch shimeji mushrooms, stems and caps; Glaze, recipe follows; 24 ½-inch pieces mochi rice cake; 1 tablespoon butter; 1 Parmesan; 1 green onion, chopped; Toasted sesame seeds as needed