Whisk together sugar, egg, canola oil, and vanilla and milk. Set aside.
Stir together baking powder, baking soda, salt and 3 cups flour. Make a well in center of flour mixture and pour liquid into well. Stir until almost combined, then add 2 cups fresh blueberries. Continue to fold ingredients together being careful to not stir too hard in order to keep blueberries intact.
Turn dough out onto well floured board and gently knead dough to form a soft dough that holds together well, adding more flour as needed. Pat dough down to about 3/4 inch thick. Cut cookies with biscuit cutter and place on parchment lined cookie sheet.
Bake at 350 for 10-12 minutes or until cookies have puffed and no indent is made when top of cookie is touched with your finger.
Remove from oven and cool at least 15 minutes.
To make the icing
Stir together powdered sugar, lemon juice and zest. Add more powdered sugar if you want a thicker glaze. Frost cookies with glaze and let sit until glaze has completely dried. Store in airtight container.