1. Remove stems from mushrooms; reserve stems. Thinly slice mushroom caps; set aside. Combine mushroom stems, Chicken Stock, and next 4 ingredients (through star anise) in a large saucepan. Bring to a boil. Cover, reduce heat, and simmer 20 minutes. Remove from heat. Let stand 10 minutes. Strain stock through a sieve over a bowl; discard solids.
2. Cook udon noodles according to package directions, omitting salt and fat. Drain and rinse with cold water; drain well.
3. Heat a large saucepan over medium-high heat. Add canola oil to pan; swirl to coat. Add reserved sliced mushroom caps to pan, and sauté for 2 minutes. Add minced ginger and minced garlic; sauté for 1 minute. Add sake, and cook for 4 minutes, scraping pan to loosen browned bits. Add stock to pan. Bring to a boil, and reduce heat to medium-low. Add chicken, soy sauce, 1 tablespoon honey, and salt; simmer for 2 minutes or until chicken is thoroughly heated. Divide noodles evenly among 4 bowls. Add 1 1/2 cups soup to each bowl. Sprinkle each serving with 1 tablespoon green onions.
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