In a mortar and pestle, mash 2 tablespoons brown sugar, 2 large chopped garlic cloves, 2 teaspoons salt, and 1 teaspoon black pepper to form a paste.
Spoon paste into a resealable plastic bag, add the 10 chicken drumsticks, coat with the paste, squeeze out excess air, and seal the bag.
Marinate the chicken in the refrigerator for 8 hours or overnight.
Heat 2 tablespoons vegetable oil in a small saucepan over medium heat.
Cook and stir 1/2 cup chopped onion in hot oil until softened, about 5 minutes.
Stir in 3/4 cup ketchup, 2 tablespoons white wine vinegar, and 2 tablespoons Worcestershire sauce into onions, bring to a simmer, and cook until flavors blend, about 10 minutes.
Preheat grill for medium heat and lightly oil the grate.
Remove chicken from the bag and discard marinade.
Cook chicken on the preheated grill until lightly browned on all sides, about 1 minute per side.
Turn off one of the grill burners or move the coals, and move chicken so there is no heat source directly below it.
Baste drumsticks with the sauce and cook for another 10 minutes, then turn and baste chicken with sauce again.
Continue to grill drumsticks until juices run clear, 10 to 15 minutes more. An instant-read thermometer inserted near the bone should read 165°F (74°C).