Heat the oil in a large Dutch oven over medium-high heat. Season the pork shoulder pieces with 1 tablespoon of the kosher salt. Brown the pork in batches for about 10 minutes, turning so all sides get a light color.

Once browned, add the chopped onion, crushed garlic, lime juice, orange juice, chili powder, dried oregano, and ground cumin. Stir for one minute to combine the flavors.

Pour in the chicken broth and bring the mixture to a boil. Reduce the heat to low, cover the pot, and let it simmer for 2 1/2 hours until the pork is very tender.

Preheat your oven to 400°F. Remove the pork from the pot and let it cool slightly. Use two forks to shred the pork, keeping some of the cooking liquid.

Spread the shredded pork on a baking sheet, drizzle with reserved liquid, and bake for about 20-30 minutes, turning once to crisp the edges (add more liquid as needed to prevent it from drying out). If needed, use the broiler setting to make it even crisper.