Bring a large pot of lightly salted water to a boil. Cook udon in boiling water, stirring occasionally, until noodles are tender yet firm to the bite, 10 to 12 minutes. Drain and rinse with cold water. Stir in a few drops of sesame oil.
Heat the remaining sesame oil in a large skillet over medium heat. Cook broccoli until bright green and still crunchy, about 5 minutes. Add green bell pepper and carrots; cook and stir until slightly softened, about 2 minutes. Add zucchini; cook until slightly softened, about 2 minutes more. Add soy sauce, mirin, chili-garlic sauce, and ginger; stir to combine. Mix in the noodles; cook and stir until noodles absorb some of the sauce, 1 to 2 minutes more.