Combine chicken stock, water, ginger, scallions, chile, dill, and reserved mushroom stems in a medium pot over medium heat. Bring to a low simmer. Add chicken, and cook until an instant-read thermometer inserted into the thickest part of the meat reaches 160 degrees, 13 to 15 minutes.
Remove chicken, and cover. Strain broth through a fine sieve; discard solids. Return broth to pot, and bring to a simmer. Add sliced mushroom caps and bok choy, and simmer until tender, about 5 minutes.
Slice chicken, and divide among 4 bowls. Ladle broth, with vegetables, over chicken, and garnish with scallions and chile.