Add the sugar and lemon zest to a large mixing bowl. Rub the lemon zest into the sugar using your fingers, until the sugar is aromatic.
Add the oil, milk, egg, lemon juice, and lemon extract to the sugar mixture. Mix using a hand mixer on medium speed until thoroughly combined. Set aside.
Combine the flour, baking powder, baking soda, and salt to a large mixing bowl. Whisk to combine.
Gradually add the dry ingredients to the wet ingredients alternately with the milk, beginning and ending with the dry ingredients until combined.
Divide the cookie dough in half. Flatten and roll out dough to approximately ¼ inch thick.
Cut into round cookie shapes using a cookie cutter.
Transfer the cookies to the parchment lined baking sheet. Bake for 12 minutes, or until cookies are lightly golden around the edges. Completely cool the cookies on a cooling rack.
Make the Icing
Combine the powdered sugar, lemon juice, and lemon zest in a medium sized mixing bowl until smooth.
Dip the tops of each cookie into the glaze and return the cookie to the cooling rack. Sprinkle the tops of the glaze with lemon zest (if using).
Allow the cookies to dry completely before storing in an airtight container.