Preheat your oven to 325°F. Pat the 3 ½ lb paleta de cerdo con hueso completely dry with paper towels — this is key for a good sear. Season all over generously with 1 ½ cucharaditas de sal kosher and 1 cucharadita de pimienta negra.

Heat the 2 cucharadas de aceite de oliva in a large Dutch oven over medium-high heat until shimmering. Add the 3 ½ lb paleta de cerdo con hueso and sear undisturbed for 4–5 minutes per side until deeply golden-brown on all sides, including the ends. Don't rush this step — the crust adds major flavor. Transfer the 3 ½ lb paleta de cerdo con hueso to a plate and set aside.

Reduce the heat to medium. Add the sliced cebolla amarilla, 3 zanahorias peladas y cortadas en trozos de 2 pulgadas, and 3 tallos de apio to the Dutch oven. Cook, stirring occasionally, until softened and lightly browned, about 5–6 minutes. Scrape up any browned bits from the bottom as you stir.

Add the smashed 4 dientes de ajo triturados and 2 cucharadas de pasta de tomate. Stir and cook for 1–2 minutes until the 2 cucharadas de pasta de tomate darkens slightly and becomes fragrant.

Pour in the 2 tazas de sidra de manzana dura seca o semiseca preferida and 1 taza de caldo de pollo. Stir in the 2 cucharadas de mostaza de grano entero. Bring to a gentle simmer and scrape up any remaining browned bits from the bottom of the pot.

Nestle the seared 3 ½ lb paleta de cerdo con hueso back into the pot — it should be about halfway submerged in liquid. Tuck in the 5 ramas de tomillo sprigs, 2 hojas de laurel, and 2 tazas de sidra de manzana dura seca o semiseca preferida wedges (if using) around the 3 ½ lb paleta de cerdo con hueso.

Cover the Dutch oven with a tight-fitting lid and transfer to the preheated oven. Braise for 2.5 to 3 hours, turning the 3 ½ lb paleta de cerdo con hueso once halfway through, until the meat is completely fork-tender and pulls apart easily. A fork should slide in with almost no resistance.

Carefully transfer the 3 ½ lb paleta de cerdo con hueso to a cutting board and tent loosely with aluminum foil to rest. Discard the 5 ramas de tomillo sprigs and 2 hojas de laurel. Using a ladle or large wooden spoon, skim off any excess fat from the surface of the braising liquid.

Place the Dutch oven on the stovetop over medium-high heat. Simmer the braising liquid and vegetables for 8–10 minutes until slightly reduced and the sauce coats a wooden spoon. Remove from heat and swirl in the cold 1 cucharada de mantequilla sin sal until melted and glossy. Add the 1 cucharadita de vinagre de sidra de manzana and taste for 1 ½ cucharaditas de sal kosher, adjusting as needed.

Use two forks or your hands to pull the 3 ½ lb paleta de cerdo con hueso into large, rustic chunks or shred it completely — your choice. Arrange on a serving platter or individual plates, spoon the pan sauce and vegetables generously over the top, and garnish with fresh cucharadas de perejil fresco de hoja plana toscamente picado if desired. Serve immediately.