Bring first 7 ingredients to a boil in a medium saucepan over medium heat. Reduce heat; simmer, stirring often, 6 minutes. Remove 1/4 cup chili sauce mixture; brush evenly on drumsticks. Reserve remaining mixture for basting.
Prepare a hot fire by piling up charcoal on one side of grill, leaving other side empty. (For gas grills, light only one side, heating to 400° to 500°.) Arrange drumsticks on food grate on unlit side.
Grill drumsticks, covered with grill lid, 1 hour or until chicken is done, turning once and basting with remaining chili sauce mixture every 15 minutes.