Heat the apple cider in a non-reactive saucepan.
In a bowl, combine the softened butter, brown sugar, ground nutmeg, cinnamon, and ground cloves. Whip until the butter becomes creamy and the ingredients are incorporated. Place the graham crackers and pumpkin pie spice in a plastic baggie and crush with a rolling pin. Combine the rum extract with the non-dairy whipped topping. In a footed coffee glass, place a single cinnamon stick and a slice of spiced butter. Pour 1 shot whiskey into the glass. Ladle the hot cider to fill the glass. Garnish with a dollop of rum-flavored topping and a sprinkle of graham cracker crumb mixture. Serve warm.