Preheat the oven to 350* and line a rimmed baking sheet with parchment paper.
Place the drumsticks in a large mixing bowl. Combine the seasonings together (asafetida, chili powder, smoked paprika, ground ginger, salt and pepper) in a small bowl and mix them thoroughly.
Sprinkle the spices over the drumsticks, and using clean hands, rub the seasonings into the drumsticks until they are thoroughly coated. Refrigerate the seasoned drumsticks for at least 30 minutes to allow the flavors to infuse.
Lay the drumsticks out on the baking sheet with a little space between each one. Drizzle them with the garlic infused olive oil. Bake for 20 minutes. Flip the drumsticks. Bake for 20 more minutes.
Remove the baking sheet from the oven, and brush each drumstick with BBQ sauce. Return to the oven and bake for 10 minutes. Remove from the oven, flip, brush with BBQ sauce and return to the oven and bake a final 10 minutes.
Remove from the oven and serve hot with extra BBQ sauce.