In a small saucepan, combine the 1/2 cup of the sugar, 1/2 cup water, lemon juice, cinnamon sticks and candied ginger. Stir over medium heat until the sugar dissolves and the syrup comes to a boil. Reduce the heat and simmer for 5 minutes. Cool to room temperature. Strain and transfer to a small bowl. (Can be made 2 days in advance). Pour 1/4 cup of the cinnamon syrup into a large pitcher. Place the remaining syrup into a shallow bowl.
Grind the gingersnaps and the remaining 1 tablespoon sugar in a mini processor to fine crumbs. (Alternatively, the cookies can be placed in a resealable plastic bag and crushed with a rolling pin or the bottom of a small saucepan.) Pour the crumbs into a second shallow bowl.
Dip the rim of each martini glass into the cinnamon syrup. Dip the moistened rims into the cookie crumbs several times to coat.
Stir the apple juice concentrate and sparkling apple cider into the pitcher. Pour into the prepared glasses and garnish with apple slices.