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Place 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.
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Combine applesauce, 1 cup sugar, and butter; beat at medium speed for 3 minutes.
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Add molasses and egg; beat well.
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Combine flours and next 5 ingredients (flours through cloves), stirring well with a whisk.
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Gradually add flour mixture to sugar mixture, beating until blended.
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Cover and freeze dough 30 minutes or until firm.
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While dough chills, preheat oven to 375°F.
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With moist hands, shape dough into 32 (1-inch) balls.
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Roll balls in 1/4 cup sugar.
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Place 3 inches apart on baking sheets coated with cooking spray.
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Bake for 8 to 10 minutes.
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Cool on pans 5 minutes before removing from pans to cool completely on wire racks.