In a medium-size mixing bowl, combine pumpkin, sugar, cinnamon, ginger, and cloves, and whisk thoroughly. Add ice cream and fold until combined. Pour into freezer container, and freeze several hours or overnight.

Place twelve frozen gingersnaps upside down on a cookie sheet. Put a scoop of ice cream on top of them, and cover with the other twelve. Gently press down on the top cookie to spread the ice cream filling. If the ice cream is too soft, put the cookie sheet in the freezer for another hour. If you worked quickly, and the ice cream is still set, individually wrap each sandwich in plastic. Store the sandwiches in a freezer bag.