Brown your butter in a skillet over medium heat until it is a deep golden brown. Set aside to cool to room temperature. When adding it to the cookie dough, it should be the texture of solid room temperature butter.
In a saucepan, add the water, blueberries, and sugar. Mix well and bring to a simmer over medium heat. While the blueberries cook down, add the cornstarch and remaining water into a small bowl. Whisk well to combine. Once the blueberries are bubbling, pour the cornstarch slurry in. Lower the heat and allow the mixture to continue cooking and thicken. This should take 3-5 minutes. Once the compote is done, let it completely cool in the refrigerator.
To prepare the cookies, add the cooled brown butter, brown sugar, and granulated sugar to the bowl of a stand mixer with the paddle attachment. Whip it until light and fluffy. Then whip in the egg, egg yolk, and vanilla extract.
Turn the mixer off and add the flour, cornstarch, baking powder, salt, and cinnamon. Slowly turn the mixer on and mix until just combined. Do not over mix at this step.
Roll the cookies into roughly 3.5 oz balls (you can make them smaller or bigger to your preference). When shaping the ball, press a divot into the top to eventually house the blueberry compote.
Once shaped, scoop 1 tbsp of the blueberry compote onto each cookie. Cover the dough balls and allow them to cool in the refrigerator or freezer overnight.
Once you are ready to bake, preheat the oven to 350°F. Then prepare the streusel by adding the brown sugar, flour, cinnamon, salt, and butter to a small bowl. Use your finger to pinch the dry ingredients into the butter. Do this until a crumbly, shaggy mixture is achieved.
Place your cookies on a baking sheet (6 per sheet for 3.5 oz dough balls). Sprinkle the streusel crumble over the cookies. Bake them for 12-15 minutes, until the centers are fully set and the edges are an even golden brown color.
Allow the cookies to cool slightly before serving to allow the berries to fully set.