In a medium bowl, mix together the sesame oil, garlic, ginger, pepper, sugar, teriyaki sauce, soy sauce and 1 cup water; add to the pot. Return all the drumsticks to the pot, and toss to coat in the liquid.
Cook, uncovered, over medium-high heat for 20 to 25 minutes, or until chicken is cooked through, stirring often. Remove chicken and set aside. (If you are not using hoisin sauce, chicken is done at this point; also remove and discard ginger slices.)
To serve at room temperature later or to transport elsewhere in a covered container: No need to add hoisin sauce, as it will get gloppy. Keep drumsticks refrigerated until shortly before serving. The chicken also can be reheated.
To serve for immediate meal: Add hoisin sauce to sauce in the pot and cook on high heat for 10 minutes, until sauce starts to thicken. Stir often to prevent burning. Add chicken and cook an additional 5 minutes, coating chicken with sauce. Remove slices of ginger and discard.
Variation: If you prefer to grill or broil the chicken, first marinate it overnight with the same ingredients, but do not add the water. Include the hoisin sauce, if available. The chicken gets much less of a garlic-ginger flavor when it's marinated this way, but it is extremely moist and tender.