Mix the paprika, garlic powder, brown sugar, dry mustard, and kosher salt together in a small bowl; spread evenly over the pork roasts. Refrigerate the seasoned roasts at least 3 hours.
Preheat oven to 300 degrees F (150 degrees C).
Place pork roasts into a large roasting pan.
Bake the roasts in the preheated oven until pork is tender and shreds easily with 2 forks, 4 to 6 hours.
Shred the pork and place into a large pot over medium heat.
Mix the meat with the barbeque sauce. Bring the mixture to a simmer and cook until the flavors have blended, about 30 minutes.
