Peel and slice the ginger into medallions. Remove any dry or black layers of the lemongrass and cut into 1 inch sections. Wash the lime well to remove any waxes or residues and remove the peel with a vegetable peeler. Try to get as little of the white pith as possible on the lime peel.
Place the ginger, lemongrass, lime peel, both cans of coconut milk, and chicken broth in a pot. Place over medium heat and stir to combine the ingredients and dissolve the coconut milk solids. Bring the liquid to a simmer.
While the soup is coming up to a liquid, thinly slice the chicken. Once the broth reaches a simmer, add the chicken, allow it to come back up to a simmer and let it cook for about ten minutes or until the chicken is fully cooked.
Once the chicken is fully cooked, turn off the burner. Add the juice from the lime and the fish sauce. Serve hot with a few sprigs of fresh cilantro. Serve over hot jasmine rice if desired.