Bring a large pot of salted water to a boil for the udon noodles. Fill a second pot with a steamer basket and 2 inches of water and bring to a boil.
Pat the chicken breast dry and sprinkle with salt and pepper. Once the water is boiling in the steamer pot, reduce to a simmer. Place the chicken in the steamer basket and cover with a lid. Steam for about 10-15 minutes. Check after 10 minutes. The chicken is done when it reaches 165°F. Remove the chicken from the pot and let cool slightly before shredding by hand or with two forks.
Another option is to poach the chicken. To poach the chicken, bring a large pot of water seasoned with the salt and pepper to a simmer. Add the chicken and simmer for 10-15 minutes. The key is to never let the water come to a boil. Check after 10 minutes and remove the chicken when it reaches 165°F. Shred the chicken as stated above.
Meanwhile, prepare the noodles by combining the brown sugar, rice wine vinegar, soy sauce and sesame oil in a small bowl. Make sure the sugar is totally dissolved.
Cook the udon noodles to al dente according to the package directions. Rinse with cool water and drain well. Toss the noodles while still warm in the sauce and put in the refrigerator until ready for use.
In a medium bowl, combine the shredded cabbage, chopped cilantro, and grated carrots.
For the peanut dressing, mix all the ingredients together using an immersion blender, food processor or a whisk.
To serve, divide the udon noodles between bowls and top with cabbage, carrot, and cilantro mixture. Top with shredded chicken. Garnish with fried shallots, crushed peanuts, and sesame seeds. Serve with a side of peanut dressing.