In a shaker, muddle the apple until the consistency of applesauce. Add the remaining ingredients and fill with ice. Shake for 10 seconds and strain into a chilled cocktail glass. The drink should have a frothy and creamy head. Garnish with an apple slice.
Add 1 tea bag and the vodka to a Mason jar or other sealable glass container and let infuse for 1 to 2 hours before removing the tea bag. In the meantime, add the cider to a small saucepan and heat over medium heat until hot but not boiling. Turn off the heat and add the other tea bag. Let steep for 3 to 5 minutes and remove the tea bag.