Deseed and chop one of the honeycrisp apples. In a small saucepan, combine the maple syrup, 3/4 cup of water, the chopped apple, 1 cinnamon stick, the allspice, and the ginger. Simmer for 12-15 minutes, stirring and mashing the apple as it softens. Strain the solids through a fine mesh sieve, squeezing the liquid from the apples as best you can. Discard the solids.
In a shaker, muddle the mint leaves with the apple syrup and lime juice. Pour in the spiced rum and apple cider. Fill with ice and shake until cold.
Pour or strain the cocktail into your favorite glass and top with a few more ice cubes as needed. Finish with a splash of sparkling water. Garnish with apple slices and a mint sprig.