To prepare the infused caramel-cinnamon syrup, combine the sugar and water in a 1.5-qt. stainless sauce pan. Over medium-high heat, whisk until the sugar is dissolved.
Continue to cook without stirring. Keep a close eye on the pan and note when the color of the syrup begins to change to amber. Heat until the syrup develops a rich, medium amber color. Turn off heat, but leave the pan on the burner.
Carefully whisk the cider into the syrup. Add the broken cinnamon sticks then let the syrup cool completely (approximately 15 minutes).
Once completely cooled, remove the cinnamon sticks with a slotted spoon. Reserve the syrup. (Makes enough syrup for 6 - 8 cocktails.)
To prepare the cocktail, pour all the ingredients into a martini shaker. Shake vigorously for 30 seconds. Strain into a chilled martini glass. Garnish with a dollop of whipped cream and a cinnamon stick or a thin slice of apple and a light dusting of cinnamon. (Makes one cocktail.)