Make the ginger syrup: Combine the sliced ginger, sugar and 1 cup water in a small saucepan over medium-high heat. Bring to a boil, stirring until the sugar dissolves; reduce the heat to medium and cook until syrupy, 15 to 20 minutes. Strain and let cool.
For each cocktail, combine 2 tablespoons each of the ginger syrup and apple juice, 1 1/2 ounces rum and 3 or 4 gratings of ginger in a cocktail shaker with ice. Shake well and pour into a short glass. Top with 1/2 cup beer and garnish with apple slices.
Photograph by James Wojcik
Recipe courtesy Chuck Hughes for Food Network Magazine