In a large bowl, combine the flour and sugar. Add the cold, cubed butter and use a pastry cutter or your hands to work it into the flour until the mixture resembles coarse crumbs.
Add the egg, vanilla extract, and salt. Mix until the dough comes together. Shape the dough into a disc, wrap it in plastic wrap, and refrigerate for at least 1 hour.
Make the Blueberry Filling:
In a saucepan, combine the blueberries, sugar, lemon juice, and cornstarch. Cook over medium heat, stirring frequently until the mixture thickens and the blueberries burst. Remove from heat and let it cool.
Preheat and Roll:
Preheat your oven to 350°F (175°C). On a floured surface, roll out the chilled cookie dough to about 1/8-inch thickness.
Cut and Fill:
Using a round cookie cutter or the rim of a glass, cut out circles from the rolled dough. Place half of the circles on a lined baking sheet.
Spoon a small amount of the blueberry filling onto the center of each circle, leaving a border around the edges.
Top and Seal:
Place another cookie dough circle on top of each filled one. Press the edges together to seal, creating a pie-like appearance.
Bake to Golden Perfection:
Bake in the preheated oven for 12-15 minutes or until the edges are golden brown. Allow the cookies to cool on the baking sheet for a few minutes before transferring them to a wire rack to cool completely.
Glaze and Enjoy:
In a small bowl, whisk together powdered sugar, milk, and vanilla extract to make the glaze. Drizzle the glaze over the cooled cookies and let it set
