In a small saucepan over medium heat, bring ginger, cinnamon, cider, honey, and cayenne (if using) to a simmer. Remove from heat, cover, and steep 10 minutes.
Using a slotted spoon, remove and discard ginger and cinnamon. Briefly reheat cider over medium heat until steaming.
Divide cider between 2 mugs. Cut lemon in half, then squeeze juice into mugs. Add bourbon and stir to combine. Garnish with apple slices (if using) and cinnamon sticks.
