In a small saucepan, mix together the blueberries, granulated sugar, cornstarch, and water. Cook over medium heat, stirring occasionally, until the mixture thickens, which should take about 5 minutes. Once thickened, remove the pan from heat, gently mash the blueberries to release more juice, and set the mixture aside to cool completely.

In a small mixing bowl, beat the cream cheese with the granulated sugar and vanilla extract until the mixture is completely smooth and creamy. Set this aside while you prepare the cookie dough.

In a large bowl, cream together the softened butter, granulated sugar, and brown sugar until the mixture is light and fluffy. Add in the egg and vanilla extract, mixing until fully incorporated. In a separate bowl, whisk together the all-purpose flour, baking soda, and salt. Gradually add the dry ingredients to the wet mixture, stirring just until a soft dough forms—do not overmix.

Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper. Scoop out dough in roughly 2-tablespoon portions and place each ball on the prepared sheet, spacing them about 2 inches apart. Use your thumb or the back of a spoon to gently press a small indentation into the center of each dough ball.

Into each indentation, spoon a small amount of the prepared cheesecake filling. Top it with a small dollop of the cooled blueberry mixture. Use a toothpick or the tip of a knife to swirl the cheesecake and blueberry fillings together slightly to create a marbled look.

Place the cookies in the oven and bake for 12 to 15 minutes, or until the edges are set and the centers still look slightly soft. Remove the cookies from the oven and allow them to rest on the baking sheet for about 5 minutes before transferring them to a wire rack to cool completely.