Remove any tough ends or hard outer husks of lemongrass, cut into 2-inch pieces, and smash with the side of a knife. Tie lemongrass in a loose bundle with cheesecloth.
Rinse chicken and place in a medium soup pot. Add 6 cups cold water to cover chicken. Season with salt and pepper. Bring to a boil and skim the surface thoroughly of any impurities that rise.
Lower the heat and add carrot, onion, garlic, ginger, and the lemongrass bundle. Adjust heat so water barely bubbles. Cover and poach chicken for 25 minutes. Lift chicken out of the pot and transfer to a plate. When it's cool enough to handle, pull meat off the bones and shred the meat into large pieces. Set aside.
Add soy sauce to broth and return to a boil. Add noodles, scallions, shiitakes, and bok choy; cook for 4 minutes, stirring occasionally, or until noodles are tender. Add chicken and heat through. Discard lemongrass bundle. Stir in the cilantro or basil before serving.