Preheat oven to 350°F (175°C) and line two baking sheets with parchment paper or silicone mats. Toss blueberries with 1 tablespoon of flour in a small bowl and set aside.
In a medium bowl, whisk together flour, baking powder, baking soda, and salt until well combined with no lumps. Set aside.
Beat softened butter and granulated sugar together with an electric mixer on medium speed for 3–4 minutes until light, fluffy, and pale in color, scraping down the sides as needed.
Add the egg, lemon juice, lemon zest, and vanilla extract to the butter mixture, then beat on medium speed for 1 minute until well blended (the mixture may look slightly curdled — that's normal).
Gradually add the dry ingredients in three additions, mixing on low speed after each just until the flour is incorporated; do not overmix. The dough should be thick and slightly sticky.
Using a rubber spatula, gently fold the flour-coated blueberries into the dough until evenly distributed, being careful not to crush the berries.
Drop rounded tablespoon-sized portions of dough onto prepared baking sheets, spacing 2 inches apart, and bake for 11–13 minutes until edges are just set and lightly golden.
Let cookies cool on the baking sheet for 5 minutes to set, then transfer to a wire rack and cool completely (about 20–30 minutes) before glazing.
Whisk together powdered sugar, 2 tablespoons lemon juice, and lemon zest until smooth; adjust with more lemon juice or powdered sugar until pourable but not too runny.
Drizzle or spoon the glaze over completely cooled cookies, then optionally garnish with a pinch of lemon zest. Allow the glaze to set for 15–20 minutes before serving.