I would pull the meat apart and add broth and a few other ingredients (Minced garlic, olive oil, cumin, oregano, salt, brown sugar, pepper) cook it till tender.
You can add barbecue sauce or Mojo Sauce (cuban citrus sauce) before serving.
Mojo Sauce
1/2 cup ex-virgin olive oil
6 cloves garlic minced
1/2 teaspoon cumin
1/2 cup orange juice
1/4 cup lime juice
1 teaspoon salt
1/2 teaspoon pepper
Heat oil, add garlic and cumin cook just till fregrant, about 30 seconds. Remove from heat, add orange juice, lime juice, salt and pepper. Return to medium heat and simmer a few minutes so flavors will blend.
Sauce can be refrigerated for 3-4 days. This sauce is also good on grilled chicken and fish.