Pour chicken stock into a saucepan over medium heat. Add cilantro roots, ginger, lemongrass, tamarind paste, garlic, and star anise pod. Bring to a gentle simmer; cook until flavors combine, about 1 hour. Cool to room temperature, 1 hour to overnight.
Place noodles in a large bowl and cover with hot water. Set aside until noodles are softened, about 15 minutes. Drain well.
Strain off the top of the cooled stock, leaving as much sediment as possible. Bring to a boil. Reduce heat to bring to a simmer. Add chicken; cook for 5 minutes. Add rice noodles, carrots, snow pea shoots, bok choy, mung bean sprouts, and daikon. Cook until chicken is no longer pink in the center, about 5 minutes more.
Serve soup with cilantro leaves, mint leaves, and chile on the side.