Combine soy sauce, brown sugar, garlic, ginger and cornstarch in a small saucepan. Add 2 Tbsp. water and bring to a boil over high heat, stirring constantly until thickened, about 2 minutes. Cool slightly and strain, discarding solids. Preheat a broiler to high and place a broiling rack 6 to 8 inches from heat source. (Alternatively, light a charcoal fire and let coals burn to a gray ash.)
Arrange drumsticks snugly on a large foil-lined rimmed baking sheet and season with salt and pepper on all sides. Pour sauce over and turn drumsticks several times to coat thoroughly. Broil or grill until chicken is browned, about 25 to 30 minutes, or until juices run clear when chicken is pierced with a knife. While cooking, turn pan around from back to front every 2 to 3 minutes and use tongs to turn drumsticks over. If liquid in baking sheet starts to scorch, add water, 1/4 cup at a time.
Remove drumsticks from pan and cool slightly before serving.