Fill a large pot with 6 cups of water and bring it to a rolling boil, which takes about 5 minutes.

Once boiling, add the chicken drumsticks, sliced ginger, and white parts of the scallions to the pot. Allow the water to return to a boil, then reduce heat to medium-low and simmer for about 10 minutes.

Turn off the heat and cover the pot tightly. Let chicken steep in the hot water for an additional 5 minutes.

Prepare an ice water bath in a large bowl. Transfer the chicken drumsticks to the ice water to cool completely for about 20 minutes.

While cooling, combine olive oil, low sodium soy sauce, coconut sugar, and about 1/2 cup of reserved cooking water in a bowl. Whisk in more sliced ginger and chopped scallions.

Remove the chicken from the ice bath, shred it onto a serving plate, and drizzle the sauce over the top.