Place bean thread noodles in a large bowl with enough warm water to cover by 1 inch; soak until softened, about 15 minutes. Drain, then cut noodles into desired lengths.

Add chicken breasts to a large pot with enough water to cover by several inches; bring to a boil. Cook until chicken is no longer pink in centers and the juices run clear, 7 to 10 minutes. An instant-read thermometer inserted into centers should read at least 165 degrees F (74 degrees C).

Transfer chicken to a cutting board; shred into strands with 2 forks. Return chicken to the pot of boiling water.

Add chicken broth; reduce heat to medium-high. Add onion, garlic, ginger, bay leaves, fish sauce, and soy sauce; season with salt and black pepper. Bring to a boil; add noodles. Cook at a boil until noodles are translucent, about 5 minutes.

Stir in bok choy; cook until leaves slightly wilt, 1 to 2 minutes.