Make the apple jam: Peel, core and cube the apples. Combine in a saucepan with the cinnamon sticks and 2 tablespoons sugar; cover and bring to a simmer over medium-low heat. Uncover and cook, stirring occasionally, until soft, about 20 minutes. Let cool.
Meanwhile, make the simple syrup: Combine 1/2 cup each sugar and water in a saucepan and bring to a boil over medium heat, stirring until the sugar dissolves. Remove from the heat and let cool.
Make the cocktail: Combine 1 teaspoon of the apple jam, 1/2 ounce simple syrup, the lime juice and whiskey in a cocktail shaker with ice. Shake vigorously and strain into a martini glass. Garnish with a dried apple slice.
Photograph by James Wojcik
Recipe courtesy Marcus Samuelsson for Food Network Magazine