Combine the cider, maple syrup, nutmeg, orange bitters, cinnamon sticks, cloves, vanilla bean and seeds and the orange half in a medium saucepot. Whisk to combine, then bring up to a boil over medium-high heat. Immediately turn the heat down to low and let simmer for 5 minutes.
Carefully remove the cinnamon sticks, cloves and orange half. With a ladle, divide the cider into 2 mugs (see Cook's Note). Place 2 donut holes on 6-inch skewers or skewer them on the cinnamon sticks. Garnish each mug with a cinnamon stick and donut skewer or dried apple slices. Serve warm.
If using vodka, divide it between the mugs, then top with the cider.