Spoon applesauce onto several layers of heavy-duty paper towels; spread to 1/2-inch thickness. Cover with additional paper towels; let stand 5 minutes. Scrape into a bowl using a rubber spatula.
Combine applesauce, 1 cup sugar, and butter; beat with a mixer at medium speed until well blended (about 3 minutes). Add molasses and egg; beat well.
Lightly spoon flours into dry measuring cups; level with a knife. Combine flours and next 5 ingredients (flours through cloves), stirring well with a whisk. Gradually add flour mixture to sugar mixture, beating until blended. Cover and freeze dough 30 minutes or until firm.
Preheat oven to 375°.
With moist hands, shape dough into 32 (1-inch) balls. Roll balls in 1/4 cup sugar. Place 3 inches apart on baking sheets coated with cooking spray. Bake at 375° for 8 to 10 minutes. Cool on pans 5 minutes. Remove from pans; cool completely on wire racks.