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Cut the pork shoulder roast into 4 pieces, each roughly 4” x 2” x 2”. Sprinkle the pork pieces with 1 ½ teaspoons salt, and then rub with the spicy brown mustard to coat. Heat 1 teaspoon of vegetable oil in the Instant Pot using Sauté” mode adjusted to high until the oil is shimmering. (Or, over medium-high heat in a stovetop pressure cooker.) Sear the pork roast in two batches, two pieces at a time. Sear each piece on 2 sides – use the largest sides – until it is well browned, about 3 minutes a side. After searing, put the pork on a platter and save for later.
Sauté the aromatics: Add the butter to the pressure cooker pot, and swirl around to start it melting. Add the onions and sprinkle with ½ teaspoon salt and the thyme. Sauté until the onion softens, about 3 minutes, stirring and scraping the browned pork bits from the bottom of the pot with a wooden spoon. Pour in the wine, bring to a simmer, and simmer for 1 minute to boil off some of the alcohol.
Everything in the pot: Pour in the chicken broth and scrape the bottom of one last time to loosen any browned bits of onion. Add the platter of pork (and any pork juices) to the pot, then scatter the carrots on top.
Pressure cook the pot roast for 45 minutes with a natural pressure release: Lock the pressure cooker lid and cook at high pressure for 45 minutes (“Pressure Cook” or “Manual” mode in an Instant Pot), or for 38 minutes in a stovetop PC. Let the pressure come down naturally, about 20 more minutes.
Defat the sauce, and serve: Gently move the pieces of pork to a serving platter with tongs or a slotted spoon. (Or both – the pork is fall apart tender at this point.) Scoop the carrots and onions into a serving bowl with a slotted spoon. Pour the remaining liquid in the pot into a fat separator and let settle for five minutes. Cut each piece of pork in half, pour a little of the defatted sauce over it, and serve, passing the rest of the sauce and the vegetables on the side.