1. Preheat Oven: Preheat your oven to 350°F (175°C). Line a baking sheet with parchment paper.
2. Cream Butter and Sugar: In a large bowl, cream together the butter and sugar until light and fluffy.
3. Add Egg and Vanilla: Beat in the egg and vanilla extract until well combined.
4. Mix Dry Ingredients: In a separate bowl, whisk together the flour, baking powder, and salt. Gradually add the dry ingredients to the wet mixture, mixing until just combined.
5. Shape Cookies: Roll the dough into small balls and place them on the prepared baking sheet. Flatten each ball slightly with the palm of your hand.
6. Bake: Bake in the preheated oven for 10-12 minutes, or until the edges are lightly golden. Allow the cookies to cool completely on a wire rack.

For the Blueberry Filling:

1. Cook Blueberries: In a small saucepan, combine the blueberries, sugar, and lemon juice. Cook over medium heat until the blueberries start to release their juices.
2. Thicken Filling: In a small bowl, mix the cornstarch and water until smooth. Add this mixture to the blueberries and stir until the filling thickens. Remove from heat and let it cool completely.

For the Lemon Frosting:

1. Mix Ingredients: In a medium bowl, beat the powdered sugar, butter, lemon juice, and lemon zest until smooth and creamy. Adjust the consistency by adding more lemon juice if needed.

Assembly:

1. Fill Cookies: Once the cookies are cool, spread a generous amount of blueberry filling on the flat side of half of the cookies.
2. Sandwich Cookies: Top with another cookie to create a sandwich.
3. Frost Cookies: Drizzle or spread the lemon frosting over the top of each cookie sandwich.
4. Serve: Allow the frosting to set before serving.