Arrange chicken in a shallow baking dish. Combine buttermilk, Tabasco, and 2 tbsp. Old Bay. Pour over chicken, cover with plastic wrap, and refrigerate overnight.
Sift together flour, remaining 3 tbsp. Old Bay, and pepper to taste into a large mixing bowl. Remove chicken from marinade (discard marinade) and dredge pieces in seasoned flour until well-coated.
Fill a large, deep skillet with oil to a depth of about 1". Heat oil over medium-high heat until very hot but not smoking. Working in batches, add chicken and fry, turning often, until brown on all sides, about 10 minutes in all. Reduce heat to medium, cover, and continue to cook, turning several times, until tender, 15–20 minutes. Remove chicken from pan and drain on paper towels.