Preheat oven to 350 degrees F. Season deer shanks with salt & pepper.
Heat oil in a 6-quart Dutch oven on stovetop burner over medium-high heat. Add shank and fry for about 5 minutes until golden brown all over.
Add onion, carrot, celery, broth, and seasonings. Mix well, cover with lid or tightly with foil, and place in the oven at 350 degrees F, undisturbed, for 3 ½ to 4 hours.
Take Dutch oven out, remove lid, and use tongs or forks to shred meat off of the shank bone. Discard the bone and any undesirable tendons or fat.
Add BBQ sauce to the meat mixture and combine well. Simmer on stovetop over medium-low heat for 10 minutes.
Serve the Deer BBQ on buns topped with pimento cheese, pickled onions, and pickles.