Prepare the Cookie Dough:

1. Preheat the Oven: Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper.
2. Mix Dry Ingredients: In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt. Set aside.
3. Cream Butter and Sugar: In a large bowl, cream together the softened butter and granulated sugar using an electric mixer until light and fluffy.
4. Add Wet Ingredients: Beat in the egg, lemon zest, lemon juice, and vanilla extract until well combined.
5. Combine Wet and Dry Ingredients: Gradually add the dry ingredients to the wet mixture, mixing until just combined. Be careful not to overmix.
6. Fold in Blueberries: Gently fold in the blueberries, making sure they are evenly distributed throughout the dough.
7. Scoop the Dough: Use a cookie scoop or tablespoon to drop dough onto the prepared baking sheet, spacing the cookies about 2 inches apart.

Bake the Cookies:

8. Bake the Cookies: Bake the cookies in the preheated oven for 10-12 minutes, or until the edges are lightly golden. The center should still be soft.
9. Cool the Cookies: Allow the cookies to cool on the baking sheet for a few minutes before transferring them to a wire rack to cool completely.

Prepare the Lemon Frosting:

10. Make the Frosting: While the cookies cool, make the lemon frosting by beating together the powdered sugar, softened butter, lemon juice, and lemon zest in a medium bowl. Add milk as needed to reach a smooth, spreadable consistency.
11. Frost the Cookies: Once the cookies are completely cooled, spread a layer of lemon frosting on top of each cookie using a butter knife or offset spatula.