In a saucepan, melt the butter over medium heat.

Incorporate the blueberries, water, granulated sugar, cornstarch, lemon juice, ground cinnamon, and salt. Stir until well combined.

Elevate to a moderate boil using medium to medium-high heat settings. Following this, reduce the heat and allow the mixture to simmer for approximately one minute, or until the consistency is notably thick.

Remove from heat and allow it to cool for a brief period.

On a floured surface, roll out the prepared pie crust to a thickness between 1/8 and 1/4 inch. Utilize a round cutter, measuring 2.5 to 3 inches, to excise as many circular segments as the dough allows (taking into account that two segments are required for one cookie).

Relocate half of the excised rounds to a baking sheet. Lightly moisten their perimeters with water to aid in sealing post filling.

Carefully place approximately 1 and 1/2 teaspoons of the blueberry filling at the center of each round, ensuring not to overextend the capacity. Overlay each with one of the reserved rounds. Seal the edges by applying pressure with a fingertip, followed by crimping with the tines of a fork for added security.

For the egg wash, whisk together the egg and the 1/4 teaspoon of water.

Generously brush the surface of each assembled cookie with the prepared egg wash to ensure an even hue post baking. If desired, garnish the surface with either sanding sugar or granulated sugar.

With precision, make incisions atop each cookie using a sharp knife.

Bake in a preheated oven set at 350℉ for a duration of 15 to 20 minutes, or until the crust adopts a golden-brown appearance.

Upon completion, transfer the cookies to a cooling rack and allow them to come to room temperature.