In a medium pot, over low heat, add all the ingredients, except the turkey. Cook stirring to combine and dissolve the salt. When salt has dissolved, remove from the heat and let cool. Add the turkey drumsticks to a large resealable plastic bag, and pour in the brine. Seal the bag well, and put into a bowl (in case of leaks). Refrigerate for 24 hours.
After 24 hours, remove the turkey, discard the brine and preheat the oven to 350 degrees F.
In a roasting pan, fitted with a rack, add the turkey drumsticks, making sure there is room between each for even cooking. Roast for 90 minutes. Remove from the oven to a serving platter and serve hot or cold.