Add the pineapple juice, lime juice, tomato paste, garlic, and jerk seasoning to the pan. Shred the chicken with two forks, then add it back in the pan. Coat the chicken with the sauce and let it cook for 2-3 minutes.
Add olive oil in a large skillet on medium high heat. When the oil is hot, add the chicken. Cook for about 3-4 minutes per side, or until it's mostly cooked through. Turn the heat down if it really starts to splatter. Remove the chicken to a plate.
Preheat oven to 350F and move the rack to the top third of the oven.