For the glaze: In a medium saucepan, combine the dark brown sugar, rum, salt, nutmeg, cinnamon, and cloves. Stir to blend and bring to a simmer over medium heat. Cook and reduce for 3 to 4 minutes to allow the sugar and other ingredients to blend together.
For the apples: Preheat the oven to 375 degrees F.
In a baking dish large enough to hold all of the apples, arrange the apples in a single layer. Put a cinnamon stick inside the cavity made from coring each apple. Pour the glaze over the apples and top each with pieces of butter. Add the cider to the bottom of the dish and place it in the center of the oven.
Cook the apples, basting them from time to time, until they are tender but not mushy, 25 to 30 minutes (smaller apples will take less time). Serve with scoops of ice cream, sour cream, or just by themselves. They are also delicious leftover.
Recipe courtesy Alex Guarnaschelli