Combine soy sauce, water, and next 4 ingredients (water through garlic) in a large Dutch oven, and bring to a boil. Remove from heat; cover and steep for 10 minutes.
Add chicken, breast-side down; bring to a boil. Cover, reduce heat, and simmer 45 minutes, turning chicken every 15 minutes.
Remove from heat; let chicken stand, breast-side down, for 20 minutes. Transfer chicken to a cutting board, reserving cooking liquid; cool completely. Strain cooking liquid through a sieve into a bowl; discard solids. Return liquid to pot; bring to a boil. Cook until reduced to 3/4 cup (about 15 minutes).
Remove chicken from bones, and cut meat into bite-sized pieces. Discard the bones. Place noodles on a platter, and top with chicken, cucumbers, green onions, and peanuts. Drizzle with the reserved cooking liquid. Serve immediately.