For the Blueberry Compote: In a small saucepan over medium heat combine the blueberries, sugar, water and lemon juice. Stir until the sugar dissolves. Cook for 6-8 minutes until the blueberries start to break down.
In a small bowl stir together the cornstarch and remaining 1 tablespoon of water to make the cornstarch slurry. Whisk into the blueberry mixture and continue cooking and whisking for 1 minute. Remove from the heat and cool completely before adding to the cookie dough.
For the Lemon Cookie Dough: Preheat your oven to 350°F and line two baking sheets with parchment paper.
In the bowl of a stand mixer, cream together the butter, granulated sugar, and lemon zest until light and fluffy, about 5 minutes. Add in the egg, egg yolk, vanilla extract, and lemon juice. Mix until fully combined.
In a separate bowl, whisk together the flour, baking powder, and salt. Gradually add the dry ingredients to the wet ingredients and mix just until combined.
Drop the cooled blueberry compote into the dough one tablespoon at a time. After each tablespoon, give the dough one very gentle fold with a spatula before adding the next scoop. Continue until all the compote is added.
Using a 1-ounce portion scoop (about 2 heaping tablespoons) drop the dough directly into the powdered sugar. The dough will be very sticky, so gently roll in the sugar until coated before placing on the prepared baking sheets about 2 inches apart. Bake for 13-14 minutes or until the edges are set.
Cool on the baking sheet for 2 minutes before transferring to a wire rack. Enjoy!