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Preheat the oven to 350°F. Lightly oil or coat two large baking sheets with nonstick cooking spray and set aside (or use parchment paper or a silicone sheet like Silpat). Whisk together the all-purpose flour, whole wheat flour, flaxseed, baking soda, ginger, cinnamon and salt in a medium bowl.
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Combine the sugar and oil in a large bowl and beat until well blended, about 1 minute. Add the egg and molasses and continue to beat until smooth. Scrape down the sides of the bowl if necessary. Gradually beat in the flour mixture until well combined.
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With your hands, roll the dough into 1¼-inch balls and place on the prepared baking sheets, about 2 inches apart. The dough is very sticky.
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Bake, 1 sheet at a time, 10 to 12 minutes, or until the cookies turn golden brown. Cool for 5 minutes on the baking sheet before transferring the cookies to a wire rack. Repeat with the remaining dough.
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To make the glaze, place the powdered sugar and milk in a bowl and stir until well combined. If the glaze is too thick, add a few drops of milk as needed. Drizzle over each cookie in a lattice or squiggle design.