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1
Heat the oven to 375F.
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2
Pat the pork dry with paper towels and sprinkle it all over with the salt.
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3
Heat the olive oil in a large Dutch oven set over medium-high heat until hot but not smoking.
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4
Add the pork and brown it on all sides, about 10 minutes total.
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5
Add the relish, barbecue sauce, and 2 cups water and bring to a simmer.
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6
Cover the pot, then transfer to the oven.
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7
Braise the pork, turning it over once, until the center is tender but not falling apart when pierced with a paring knife, 2 1/2 to 3 hours.
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8
Transfer the pork to a cutting board and tent loosely with foil.
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9
Strain the braising liquid through a sieve into a bowl and discard the solids.
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10
Skim off any fat that collects on the surface and cover the bowl to keep the liquid warm.
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11
Cut the meat across the grain into 6 slices (each slice may break into 2 or 3 smaller pieces, depending on the part of the shoulder you have; discard any kitchen string if necessary).
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12
Heat the vegetable oil in a 12-inch heavy nonstick skillet set over medium-high heat until hot but not smoking, then sear the pork slices in 2 batches, turning over once, until browned, about 3 minutes per batch.
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13
Transfer the pork to plates and spoon some braising liquid over the top.
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14
Pass the remaining braising liquid at the table.
