Place chicken, breast-side down, in a large pot or Dutch oven; add water to cover. Bring to boil over medium-high heat; reduce heat and let simmer 10 minutes. Skim off and discard foam. Add next 10 ingredients (carrots through bay leaf); cover and simmer 5 minutes. Turn heat off; cover and let stand undisturbed for 45 minutes.
Carefully transfer chicken to a large baking dish or carving board (cutting board with a groove to catch liquid) until cool enough to handle.
Meanwhile, strain solids out of broth. Rinse out pot and return strained broth; stir in kosher salt (plus more to taste) and bring to a low simmer.
Bring a separate pot of water to boil for the noodles.
Remove and discard skin and bones from chicken and shred into meat into bite-size pieces.
Place shredded chicken in a small bowl; add soy sauce and pepper and toss to coat.
Cook noodles according to package directions; drain and divide evenly between 6 serving bowls. Top each bowl with hot broth, seasoned chicken and garnish as desired. Serve immediately.