In a shallow container, whisk together mayo, balsamic vinegar, thyme, salt and pepper.
Place the drumsticks in the container in a single layer, making sure all sides are coated with the marinade.
Cover and refrigerate for 6-24 hours, flipping them halfway through.
Take out the drumsticks 30 minutes before cooking them.
Preheat the grill to medium high.
Place drumsticks on the grill and turn them every 5 minutes, brushing on a layer of marinade every time you turn.
Make sure all sides are cooked through, and take them off after about 25 minutes.
Let the drumsticks rest for 5-10 minutes before serving.