Wash the orange well, and use the vegetable peeler to remove long strips of zest (the orange part, not the white part underneath) from half of it. Slice the other half of the orange into thin half moons.
Put the cider and cranberry juice in the pot, and add the orange zest, the cinnamon sticks, the cloves, the sugar, and a tiny pinch of salt.
Put the pot on the stove, turn the heat to medium-high, and heat the mixture until it’s a bit steamy, about 7 minutes. Turn the heat down to low and simmer the mixture for 30 minutes.
Set the sieve over the heatproof container, strain the cider mixture, and throw away the solids in the sieve. Add the vanilla and stir to mix. Pour the mixture into mugs, float an orange slice in each one if desired, and serve.