Put drumsticks in a large bowl with enough water to cover. Stir salt into water to dissolve. Brine drumsticks in the refrigerator for 2 hours; drain. Rinse drumsticks with clean water and pat dry.
Preheat the oven to 350 degrees F (175 degrees C).
Pour broth into a roasting pan. Brush melted butter over drumsticks to coat; season with black pepper. Lay drumsticks in broth. Cover the roasting pan with a lid.
Bake in the preheated oven until no longer pink at the bone and the juices run clear, 1 hour 45 minutes to 2 hours. An instant-read thermometer inserted into the thickest part of the drumstick near the bone should read 165 degrees F (74 degrees C).