Use a serrated knife to cut away the pineapple skin. Slice the pineapple into ½-inch-thick rings (I usually get 8 rings.) Use a round cutter to remove the core from each ring.
Melt salted butter in a large rimmed pan over medium heat. Stir in brown sugar and vanilla until smooth. Bring mixture to a simmer. Add a single layer of 4 pineapple rings and cook for 4 minutes at a low boil. Flip rings over and continue cooking until tender, about 4 minutes more. Use a slotted spatula to transfer caramelized pineapples to a rimmed serving dish. Cook remaining pineapple rings.
Pour caramel sauce remaining in the pan over the pineapple rings to serve. Leftovers can be stored covered in the refrigerator.