In a large pot over medium-high heat, combine tomato sauce, granulated sugar, brown sugar, soy sauce, vinegar, and mustard. Stir well and bring the mixture to a gentle boil.
Add the chicken drumsticks to the pot. Reduce the heat to medium-low, cover, and simmer for 20-30 minutes, or until the chicken is just cooked through. Once done, transfer the chicken to a plate and set aside some sauce in a cup.
Preheat and grease the grill grates with oil. Heat the grill to high heat. Place the chicken drumsticks on the grill. The chicken is already cooked, so you're just looking to char the skin. Brush some extra BBQ sauce on the chicken as you grill, rotating the drumsticks until lightly charred on each side. This should only take a few minutes if the grill is hot enough.
Serve the BBQ chicken drumsticks with your favorite BBQ side dishes.
Notes
Notes: You can adjust the sweetness or tanginess of the BBQ sauce by tweaking the sugar and vinegar proportions to suit your taste.