Preheat the oven to 400 degrees F (200 degrees C). Line a baking dish with foil.

Whisk egg in a shallow bowl. Mix Parmesan, pepper, and salt together in another shallow bowl. Dip each chicken drumstick in egg; lift up so excess egg drips back into the bowl. Roll in Parmesan mixture until well coated. Place into the prepared baking pan.

Bake in the preheated oven until golden brown on the outside, no longer pink at the bone, and the juices run clear, about 45 minutes. An instant-read thermometer inserted near the bone should read at least 165 degrees F (74 degrees C).