Start by preparing the pork shoulder. Mix sea salt, smoked paprika, garlic powder, onion powder, and ground black pepper in a small bowl. Rub this mixture all over the pork shoulder evenly.
Place the seasoned pork into a slow cooker. Pour apple cider vinegar and chicken broth over the pork. Set the slow cooker to low and cook for about 4 hours, or until the pork is tender enough to be easily shredded with a fork.
While the pork is cooking, prepare the keto BBQ sauce. In a saucepan, combine tomato paste, apple cider vinegar, Worcestershire sauce, liquid smoke, powdered erythritol, sea salt, black pepper, smoked paprika, cayenne pepper (if using), and water. Stir well to combine.
Bring the sauce to a simmer over medium heat, then reduce the heat and let it simmer for about 20 minutes, stirring occasionally, until the sauce thickens and the flavors meld together. Once done, set aside.
Once the pork is cooked, transfer it to a large bowl and shred it using two forks. Pour the homemade keto BBQ sauce over the shredded pork and mix well to combine.
To assemble the burgers, place a lettuce leaf on each plate. Top with a generous serving of the BBQ pulled pork, followed by sliced red onion and a slice of cheddar cheese if desired.
Serve immediately, with extra BBQ sauce on the side if you like.