Combine chicken, broth, sliced ginger, crushed garlic, green onions, and allspice in a large stock pot. Bring to a boil. Cover, reduce heat, and simmer 20 minutes. Remove from heat and let stand 10 minutes. Remove chicken and set aside. Strain stock over a bowl; discard solids.
Using your hands or a fork, shred chicken and set aside.
Cook udon noodles according to package directions, omitting salt and fat. Drain and rinse with cold water.
Heat large stockpot over medium-high heat. Add oil and swirl to coat. When hot, add carrots and broccoli and sauté 3 minutes. Add mushrooms and sauté another 2 minutes. Add minced ginger and garlic; sauté for 1 minute. Add mirin and wine, and cook for 4 minutes, scraping pan to loosen browned bits. Add reserved stock to pot; bring to a boil, and reduce heat to medium-low. Add bok choy, shredded chicken, soy sauce, honey, and salt; simmer for 2 to 5 minutes, until chicken is thoroughly heated and bok choy is wilted. Season with fresh pepper and more salt if needed.
To serve, place about 1 cup (3-4 ounces) cooked udon noodles in a bowl. Add about 2 cups of soup, chicken, and vegetables to the bowl. Sprinkle each serving with 1 tablespoon green onions.