Bring a large pot of lightly salted water to a boil. Add carrots and cook uncovered until tender yet firm to the bite, about 2 minutes. Drain in a colander and immediately immerse in ice water for several minutes to stop the cooking process. Drain. Repeat with celery, cabbage, chopped onion, and snow peas, cooking separately.
Heat olive oil and sesame oil in a stockpot over medium heat; cook and stir mushrooms, garlic, ginger, salt, and pepper until fragrant, 2 to 3 minutes. Add chicken broth and noodles; bring to a boil and cook for 3 minutes.
Mix carrots, celery, cabbage, onion, snow peas, and water chestnuts into broth mixture; stir in yakisoba sauce, soy sauce, lemon juice, sambal sauce, and balsamic vinegar. Simmer soup until flavors have combined, 5 to 10 minutes.
Ladle soup into bowls and top with sliced onions and bean sprouts.