Place 1/3 cup flaxseed in a blender or clean coffee grinder, and process until ground to measure 1/2 cup meal. Lightly spoon the flour into dry measuring cups, and level with a knife. Combine flaxseed meal, flour, and the next 5 ingredients (flour through ginger); stir with a whisk. Combine 1 cup sugar, apple butter, molasses, oil, and egg white, and stir with a whisk. Add to flour mixture, stirring just until moist. Cover bowl with a sheet of heavy-duty plastic wrap; place bowl in freezer 1 hour.
Preheat oven to 350°.
Lightly coat hands with cooking spray. Shape dough into 36 balls, about 1 tablespoon each. Combine 1/3 cup sugar and 1 tablespoon flaxseed in a small bowl; roll balls in the sugar mixture. Place 2 inches apart on baking sheets coated with cooking spray. Bake at 350° for 13 minutes or until golden. Cool for 5 minutes on pans. Remove the cookies from pans, and cool completely on wire racks.
Note: This cookie dough freezes well. Make dough and shape into a large ball; freeze the ball in a zip-top plastic freezer bag. When ready to use, thaw dough completely in refrigerator. Sharpe, coat, and bake as instructed.