In a large bowl, whisk together the vinegar, honey, brown sugar, rosemary, olive oil, mustard, and grlic. Add the drumsticks to the bowl and toss to coat well. Cover and let marinate in the refrigerator for at least 1 hour and up to 8 hours, tossing a few times.

Preheat the oven to 375F. Coat a baking sheet with cooking spray, line with aluminum foil, and coat again.

Remove the drumsticks from the marinade and arrange on the prepared pan. Bake, turning once about halfway through, until an instant-read thermometer inserted into the meatiest part of a leg away from the bone registers 165F, about 40 minutes. Serve right away.