In a large saucepan, cook mushrooms and garlic in hot oil over medium heat for 4 minutes, stirring occasionally. Stir in broth, vinegar, soy sauce, and red pepper or chili oil; bring to boiling. Stir in chicken and coleslaw mix or napa cabbage; reduce heat. Simmer, uncovered, for 5 minutes.
In a small bowl, combine the cold water and cornstarch. Stir into soup; simmer about 2 minutes or until slightly thickened. Remove from heat; stir in sesame oil. If desired, garnish individual servings with carrot triangles. Makes 4 servings.