Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper.
In a medium bowl, whisk together the flour, baking powder, baking soda, and salt. Set aside.
In a large bowl, cream together the softened butter and granulated sugar until light and fluffy, about 3-4 minutes.
Add the egg, vanilla extract, and lemon zest to the butter mixture, mixing until well combined.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Gently fold in the blueberries.
Drop rounded tablespoons of dough onto the prepared baking sheet, spacing them about 2 inches apart.
Bake for 12-15 minutes, or until the edges are lightly golden. Remove from the oven and let cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.
While the cookies are cooling, prepare the frosting by whisking together the powdered sugar, lemon juice, and lemon zest until smooth.
Once the cookies are completely cooled, drizzle or spread the frosting over the tops.