Vegetable Udon Stir Fry

Name: Vegetable Udon Stir Fry
Ingredients: 400 grams (14 oz) of fresh udon noodles; 1 tablespoon oil; 2 cloves of garlic, minced; 1 tablespoon minced ginger; 2 green onions, sliced and whites separated from greens; 150 grams (5 oz) broccoli florets; 150 grams (5 oz) fresh shiitake mushrooms, sliced; Half a red pepper, sliced; 1/4 cup soy sauce; 2 tablespoons rice vinegar; 2 tablespoons brown sugar; 2 tablespoons water, plus more as needed for steaming the veggies; 1 tablespoon cornstarch; Sesame seeds, for garnish