Whisk together flour, baking soda, salt, ginger, cinnamon, and nutmeg in a medium bowl; set aside. Put shortening and brown sugar in the bowl of an electric mixer fitted with the paddle attachment. Mix on medium-high speed until fluffy, about 1 minute.
Mix in egg. Reduce speed to medium. Mix in vinegar and molasses. Reduce speed to low. Gradually mix in flour mixture until soft dough forms. Let stand at room temperature until slightly firm, 10 minutes.
Divide dough into thirds. Roll 1 piece between 2 sheets of parchment paper to just over 1/8 inch thick.Transfer parchment and dough to baking sheet. Repeat with remaining dough, stacking them on the same baking sheet. Refrigerate until firm, about 3 hours (or chill in freezer 1 hour).
Preheat oven to 400 degrees. Transfer 1 sheet of dough on parchment to a lightly floured work surface. Peel off 1 layer of parchment, and then pat it back into place. Flip dough; peel away second sheet of parchment, and discard. Cut out shapes using a 4-inch gingerbread-man cookie cutter, transferring to parchment-lined baking sheets as you work. Roll out scraps; repeat.
Bake cookies until edges are just browned, about 7 minutes. While cookies are warm, poke eyes and noses on half of them using the blunt end of a skewer. Let cool on sheets on wire racks. Repeat with remaining dough and scraps.
Decorate the cookies that have faces with candies, using a small amount of melted chocolate to adhere them. Let ice cream stand at room temperature until slightly softened, about 10 minutes. Spread 1/2-inch-thick layer of ice cream onto the underside of an undecorated cookie. Place a decorated cookie on top. Smooth edges using a knife or small spatula. Transfer to a baking sheet in freezer. Repeat with remaining cookies and ice cream. Freeze until set, about 1 hour, before serving.