Preheat your oven to 300°F (150°C).
In a small bowl, combine the sea salt, black pepper, smoked paprika, garlic powder, and onion powder to create a dry rub.
Pat the pork shoulder dry with paper towels and rub the spice mixture all over the meat, ensuring it is fully coated.
Heat the olive oil in a large Dutch oven over medium-high heat. Once hot, sear the pork shoulder on all sides until a golden-brown crust forms, approximately 2-3 minutes per side.
Pour the chicken broth into the Dutch oven, scraping any browned bits off the bottom of the pot. This will add extra flavor to the pulled pork.
Cover the Dutch oven with its lid and place it in the preheated oven. Cook for about 4 hours, or until the pork is tender enough to easily shred with a fork.
While the pork is cooking, prepare the smoky BBQ sauce. In a saucepan over medium heat, combine the tomato sauce, apple cider vinegar, Worcestershire sauce, liquid smoke, smoked paprika, garlic powder, and onion powder. Stir well and bring to a simmer.
Reduce the heat to low and let the sauce simmer for about 15 minutes, allowing the flavors to meld. Taste and adjust seasoning with sweetener, salt, and pepper as needed. Remove from heat and set aside.
Once the pork is done, remove it from the oven and let it rest for about 15 minutes. Then, using two forks, shred the pork into bite-sized pieces.
Toss the shredded pork with the smoky BBQ sauce until well coated. Serve hot.