Preheat oven to 325.
Pat chicken dry.
In Dutch oven, heat a drizzle of olive oil over medium heat. Cook the onion and garlic for 5 minutes until quite soft. Add the paprika and stir. Add the chicken and stir. Stir in salt, pepper, and BBQ sauce and bring to a simmer.
Cover the pot and bake for 90 minutes until chicken is very tender. (Alternatively, cook in a slow cooker for 5-7 hours on LOW).
Remove the chicken pieces using a slotted spoon. Use 2 forks to finely shred the chicken.
Meanwhile, put the remaining sauce, uncovered, back on the stovetop over high heat. Bring to a boil and reduce the liquid to about half its volume. Pour this thickened sauce over the shredded chicken and stir.
Serve with buns and extra BBQ sauce on the side. Refrigerate for 3 days or freeze for up to 3 months.