Marinate chicken: Pat dry the chicken drumsticks with a paper towel and sprinkle generously on both sides with kosher salt. Using a silicon brush, brush the chicken drumsticks with 1/2 cup of barbecue sauce. Set aside the rest.
Cover, and set in the refrigerator to marinate for at least 30 minutes or as long as overnight.
Sear chicken: Set the broiler to high heat with a rack positioned a few inches below it. Line a baking sheet with aluminum foil and place a metal cooling rack inside of it. Arrange the marinated chicken on the rack with space between each drumstick.
Slide the pan under the broiler, and broil chicken until you begin to see brown charred spots, about 6 minutes. Flip the chicken, and sear the other side until it is browned, about 3 to 5 minutes.
Turn the oven to temperature down to 350 degrees Fahrenheit, and move the rack into the middle of the oven.
Finish cooking chicken: Brush chicken all over with another 1/2 cup (but leave behind 1 tablespoon) of the sauce. Slide into the oven and bake about 15 minutes. Flip the chicken, and brush with the remaining tablespoon of sauce anywhere it's bare.
Continue baking until the internal temperature of the chicken reaches 165°F, approximately another 15 to 20 minutes.
Serve hot with leftover barbecue sauce. Enjoy!