Add all of the ingredients to a medium (5 to 6-quart) heavy bottomed soup pot or dutch oven, pouring in the chicken broth last to avoid splashing. Cover the pot, then place it on the stove over medium heat.
When the soup comes to a boil, turn down to low and let simmer, covered, until drumsticks are tender and the meat is just beginning to fall off the bones, about 1 hour and 15 minutes.
Remove the drumsticks with a slotted spoon and allow them to rest just until cool enough to handle.
Remove the meat from the drumsticks, discarding the bones, skin, and cartilage. Return the meat to the pot with the other ingredients.
Check for seasoning and salt to taste, if desired. Ladle into bowls and serve.