In a small saucepan (use a small saucepan for this step), combine the blueberries, ½ cup granulated sugar, and 2 tablespoons of water. Stir until combined. Let it come to a boil, then reduce heat to simmer until the blueberries open and create a sauce.
Make a slurry by mixing the cornstarch with ⅓ cup water. Ensure your blueberry mixture is hot, then add the slurry. Stir until it thickens and darkens, indicating that the cornstarch has been fully cooked out.
Set aside to cool to room temperature. For faster cooling, you can refrigerate it.
For the Blueberry Cookies:
In a medium-sized saucepan, brown 1 cup of unsalted butter over medium heat until it starts making crackling noises. Reduce heat to medium-low, stir occasionally until browned. Let it cool to room temperature.
In a separate mixing bowl, whisk together both flours, baking powder, salt, cornstarch, and cinnamon.
In a stand mixer fitted with a paddle attachment, beat the cooled browned butter, light brown sugar, and granulated sugar on medium-high speed until light and fluffy (about 3 minutes).
Beat in the egg, egg yolk, and vanilla for another 2 minutes.
On low speed, add the dry ingredients to the wet ingredients just until no more signs of flour remain. Avoid overmixing.
Gently fold in the blueberries with a rubber spatula.
Scoop out the dough into balls (approximately 3 ounces each), then press the back of a spoon into the center of each cookie to create a divot. Add the cooled blueberry compote into each divot.
Refrigerate the cookies for 2 hours or overnight before baking to solidify the butter.
For the topping
In a mixing bowl, whisk together the brown sugar, salt, cinnamon, and flour. Add the room temperature unsalted butter and combine everything with your hands until it has a shaggy look. Store it in an airtight container in the fridge for up to 7 days if prepared in advance.
Assembly and Baking:
Preheat the oven to 350°F (180°C). Line two cookie sheets with parchment paper and place 4 balls of cookie dough on each cookie sheet. Sprinkle 1-2 tablespoons of streusel on top of each cookie dough ball.
Bake for 13-15 minutes. Allow the cookies to slightly cool on the baking sheets before transferring them to a wire cooling rack to finish cooling.