Preheat the oven's broiler and set the oven rack about 8 inches from the heat source.
Pour the beer into a deep, broiler-safe baking pan. Cut half of the lime into thin slices and place into the beer. Squeeze the other half of the lime into the beer, and place the chicken legs on top. Sprinkle with coarse salt to taste.
Broil for 15 minutes, then turn the legs over, and continue broiling until the chicken is no longer pink at the bone and the juices run clear, about 15 minutes more. An instant-read thermometer inserted near the bone should read at least 165 degrees F (74 degrees C).