Position oven rack about 8 inches from heat source. Preheat broiler; line a broiling pan with foil. Pat drumsticks dry and trim off fat. Arrange in a single layer on prepared pan.
Melt butter with garlic and lemon juice over medium-low heat. Season drumsticks with salt and pepper. Brush each lightly with lemon butter.
Broil, turning often and brushing with lemon butter, until chicken is browned and crisp and has no traces of pink juices when pierced, about 30 minutes. Cool slightly. Serve drumsticks with dipping sauces.