Make the caramel syrup by heating sugar in a small saucepan over medium heat until melted and amber in color, then carefully whisk in water. Let cool completely.
In a small skillet, melt butter over medium-high heat and add the apple slice. Cook for 2-3 minutes per side until caramelized and golden. Set aside.
In an old fashioned glass, combine bourbon, caramel syrup, apple cider, and bitters.
Add a large ice cube and stir gently for 30 seconds to chill and dilute slightly.
Garnish with the caramelized apple slice and cinnamon stick. Serve immediately.