To prepare cookies, lightly spoon flour into dry measuring cups; level with a knife. Combine flour, baking soda, ground ginger, baking powder, and salt, stirring with a whisk. Place 1/2 cup granulated sugar, brown sugar, and butter in a large bowl; beat with a mixer at medium speed 3 minutes or until mixture resembles coarse meal. Add vanilla and egg; beat well. Add flour mixture to sugar mixture; beat at low speed until well blended. Divide dough in half; gently press each portion into a 4-inch circle on plastic wrap. Cover and chill 1 1/2 hours.
Preheat oven to 350°.
Unwrap each dough portion and place on 2 sheets of overlapping plastic wrap. Cover each portion with 2 additional sheets of overlapping plastic wrap. Roll, still covered, to a 1/4-inch thickness. Place dough in freezer 30 minutes or until plastic wrap can be easily removed. Remove plastic wrap; cut dough with a 3-inch gingerbread man cookie cutter to form 12 cookies. Place cookies 1 inch apart on baking sheets coated with cooking spray. Sprinkle cookies evenly with 1/2 teaspoon granulated sugar. Bake at 350° for 12 minutes or until set. Remove cookies from baking sheets; cool completely on wire racks. Repeat procedure with remaining dough and 1/2 teaspoon granulated sugar.
To prepare filling, combine ice cream and crystallized ginger, stirring well. Spread 1 1/2 tablespoons ice cream on the flat side of each of 12 cookies; top with remaining cookies, flat sides down, pressing gently. Wrap each sandwich tightly in plastic wrap; freeze 4 hours or until firm.