Preheat barbecue to medium. Season drumsticks with salt and pepper. Grill with lid closed, turning occasionally until cooked through, about 22 min.
Spread breadcrumbs evenly over a baking sheet and toast on top rack of barbecue until golden brown, 3 to 4 min. Remove sheet from grill, sprinkle evenly with cheese and herbs, then set aside to cool.
Toss drumsticks in a bowl with barbecue sauce until coated evenly. Roll in breadcrumb mixture, covering completely. Transfer to a platter and serve.
These drumsticks get even crunchier when reheated, so double the recipe for delicious leftovers.
NO BBQ? Preheat oven to 375 °F (190 °C). Spread drumsticks evenly on a parchment paper-lined roasting pan. Roast, turning occasionally, about 30 min. Meanwhile, spread breadcrumbs evenly over a baking sheet and toast until golden brown, 8 to 10 min. Sprinkle with cheese and herbs, transfer to a large mixing bowl and set aside to cool. Brush drumsticks with barbecue sauce. Roast to lightly caramelize sauce, 2 to 3 min. toss drumsticks in breadcrumb mixture.