In a 4- or 5-quart Dutch oven or stock pot, combine the broth, onion, carrots, water, soy sauce, serrano pepper, garlic, and ginger. Bring to boiling; reduce heat. Add noodles and simmer, covered, until noodles are tender, about 7 minutes, stirring occasionally.
Stir in cabbage, mushrooms, zucchini, and sugar snap peas. Return just to boiling; reduce heat. Simmer, uncovered, about 5 minutes or until vegetables are crisp-tender, stirring occasionally.
In a small bowl, stir together the cornstarch and the 1 tablespoon cold water; stir into broth mixture. Cook and stir until thickened and bubbly; cook and stir 2 minutes more.
To serve, spoon into bowls. Garnish with sliced green onions before serving.