Pat the pork pieces dry with paper towels. Season evenly with the 2 tsp kosher salt and 1 tsp black pepper. Lightly dredge each piece in the 3 Tbs flour, shaking off excess; this will help create a light crust and thicken the braising liquid.

Brown pork

Heat a large heavy-bottomed Dutch oven or ovenproof pot over medium-high heat. Add 2 Tbs vegetable oil and heat until shimmering. Working in batches if needed to avoid crowding, add pork pieces and sear until deeply browned on all sides, about 3–5 minutes per side. Transfer browned pork to a plate and set aside.

Sauté aromatics

Reduce heat to medium. Add the sliced onion, carrots, and celery to the pot with a pinch of salt. Sauté, scraping up browned bits, until vegetables are softened and beginning to brown, about 6–8 minutes. Add the smashed garlic and cook 30–60 seconds until fragrant.

Build braising liquid

Stir in 2 Tbs tomato paste and cook 1–2 minutes to remove raw taste. Add 3 fl oz dry white wine to deglaze the pot, scraping up any stuck bits from the bottom; simmer 2–3 minutes until wine reduces by about half. Return the pork and any accumulated juices to the pot. Pour in 3 cups chicken stock so the liquid reaches about two-thirds up the sides of the pork pieces (do not necessarily cover completely). Add 2 bay leaves, 3 thyme sprigs, 1 Tbs soy sauce, and 1 Tbs brown sugar. Bring the liquid to a gentle simmer on the stovetop.

Oven braise

Cover the pot with a tight-fitting lid and transfer to a preheated oven at 325°F (160°C). Braise until the pork is fork-tender and easily pulls apart, about 2 to 2 1/2 hours depending on piece size. Check once at about 90 minutes: if liquid is rapidly evaporating, add up to 1/2 cup more stock. When done, the internal temperature for very tender pork will be above 185°F (85°C) and a piece should shred with little resistance.

Finishing & Optional Garnish

Reduce and finish sauce

Carefully remove pork pieces to a cutting board or bowl and cover loosely with foil to keep warm. Place the Dutch oven back on the stovetop over medium-high heat. Skim any excess fat from the surface of the braising liquid if desired. Simmer the liquid until it reduces and slightly thickens, about 8–12 minutes, stirring and scraping to incorporate browned bits. Stir in 1 Tbs butter to enrich the sauce and adjust seasoning with salt and pepper to taste. If you want brightness, stir in 1 tsp fresh lemon juice.

Serve

Return pork to the pot to coat in the sauce or shred the pork with two forks and stir into the sauce to combine. Sprinkle with 2 Tbs chopped fresh parsley before serving. Serve with mashed potatoes, polenta, or rice and spoon extra braising sauce over the top.