Preheat oven to 400°F. Prepare one or two rimmed baking sheets by lining them with foil and setting oven-safe oven or cooling racks into them.
In a large bowl, toss the large chicken drumsticks with the vegetable oil, salt, and pepper so that all the pieces are fully coated. Divide the chicken pieces across the prepared baking sheets.
Bake the chicken pieces 50-60 minutes, until crispy and cooked through to 165°F. You can test for doneness with a thermometer inserted into the thickest part of the meat, avoiding the bone.
As the chicken is cooking, make the hot sauce. Mix the warm melted butter, cayenne pepper, black pepper, and salt in a small bowl or spouted measuring cup, and let stand for 5 minutes. Whisk in the hot pepper sauce and keep warm.
Once the chicken is fully cooked, toss the pieces with the prepared hot sauce in a large bowl until completely coated in spicy deliciousness. Maybe leave a few plain if you know some of your guests don’t care for the heat. Serve on a platter immediately with a side of homemade blue cheese dressing and a heaping pile of napkins.
The hot sauce can be made one week ahead of time. Let it cool, refrigerate, and rewarm before coating the chicken.