In a small saucepan over medium heat, combine blueberries, granulated sugar, cornstarch, water, and lemon juice. Stir frequently for 5-7 minutes until the mixture thickens to form a delicious filling. Remove from heat and allow it to cool.

Preheat your oven to 350°F (175°C) and prepare a baking sheet by lining it with parchment paper.

In a medium bowl, whisk together flour, baking soda, and salt. In a separate larger bowl, cream the softened butter and granulated sugar until fluffy, then beat in the egg and vanilla extract until smooth.

Gradually incorporate the flour mixture into the butter mixture, followed by milk, until well combined.

Use a spoon to drop dollops of cookie dough onto the baking sheet, ensuring they are spaced about 2 inches apart. Press a well into the center of each cookie dough lump.

Spoon a small amount of the cooled blueberry filling into each well, letting it nestle comfortably within the cookie base.

Place the cookies in the oven and bake for 10-12 minutes, or until the edges turn golden brown. Allow them to cool on the baking sheet for 5 minutes before transferring to a wire rack.

To prepare the icing, mix together powdered sugar, milk, and vanilla extract in a small bowl until smooth. Drizzle the icing over the cooled cookies for that final touch.