Make the BBQ sauce: Combine tomato sauce, granulated sugar, light brown sugar, soy sauce, white vinegar, and mustard in a large pot over medium-high heat; stir to combine.
Bring the sauce to a gentle boil, then reduce heat to medium-low and simmer briefly to meld flavors, stirring occasionally.
Add the chicken drumsticks to the pot with the sauce, reduce heat to medium-low, cover, and simmer for 20–30 minutes until the chicken is just cooked through.
Remove the cooked drumsticks to a plate and reserve some sauce in a cup for basting.
Prepare the grill: clean and oil the grates, then heat the grill to high.
Grill the drumsticks over high heat just to char the skin, about a few minutes per side, basting with reserved sauce and turning as each side becomes lightly charred.
Remove from grill and serve with your favorite BBQ side dishes.