Hot Buttered Apple Cider

In a small bowl, combine butter, brown sugar, cinnamon, nutmeg, cloves and salt to make spiced butter. Use a plastic spatula to transfer the butter mixture to the middle of a 12-by-12-inch piece of plastic wrap. Fold the plastic wrap over the butter mixture and shape into a log, twisting ends to seal closed. Refrigerate until ready to use. Bring the apple cider to a simmer over medium heat in a large pot, then turn the heat down to low. Remove the spiced butter from the plastic wrap and cut into 1-inch pieces. Add to the hot cider and stir continuously until all the butter is melted. Ladle the buttered cider into mugs. Top off each drink with a shot of rum and garnish with a cinnamon stick, if desired. Serve hot.