Pat the pork dry and sprinkle with salt. Heat the olive oil in a large Dutch oven over medium high heat. Add the pork and brown on all sides. Add the relish, barbecue sauce, and 2 cups of water and bring to a simmer. Cover the pot and transfer to the oven. Braise the pork, turning over once, until the center is tender but not falling apart when pierced with a paring knife, about 2 1/2 to 3 hours.
Transfer the pork to a cutting board and tent with foil.
Strain the braising liquid through a sieve into a bowl and discard the solids. Skim the fat that collects on the surface and cover the bowl to keep the liquid warm.
Cut the meat across the grain into 6 slices. Heat the vegetable oil in a 12 inch heavy nonstick skillet over medium high heat. Sear the pork slices in batches, until browned on each side.
Transfer the pork to plates and spoon some braising liquid over the top.