Make the blueberry sauce. In a medium pot over medium-low heat, combine the fresh blueberries, sugar, and lemon juice. Cook, stirring frequently, until the blueberries break down and the mixture thickens. This will take about 5-8 minutes. Set aside to cool.
Combine the dry ingredients. In a large bowl, combine the flour, cornstarch, baking powder, and salt. Set aside.
Cream the butter and sugar. In another large bowl, cream the butter and 1¼ cups of white granulated sugar until light and fluffy.
Add the blueberry sauce. Scoop 3 tablespoons of the cooled blueberry sauce out of the pot and mix it into the butter and sugar. (If you have extra sauce, put it in a mason jar for later. You can use it on pancakes.)
Add remaining liquid ingredients. Add milk, egg, and dye. Mix until just combined.
Add dry ingredients. Mix until everything is well combined and there are no dry bits of batter.
Chill the dough. Cover your bowl with saran wrap and chill in the refrigerator for 3 hours.
Prep your tools. Preheat your oven to 325ºF. Line a baking sheet with parchment paper or a Silpat. Set aside.
Pour sugar. Place the remaining 1/4 cup of sugar in a shallow plate. Set aside.
Form the cookies. Scoop the chilled cookie dough and roll it into balls, using about 1 heaping tablespoon of dough per cookie.
Roll in sugar. Place the rolled cookie dough into the sugar and roll it around to coat. Place the sugar cookie dough ball onto your prepared baking sheet. You will be able to fit 6 cookies on one standard baking sheet. (There needs to be space between the balls of dough.)
Chill the extra dough. Once you have shaped your 6 cookies, place the cookie dough back in the fridge until you are ready to shape the next batch. Or, if you have two cookie sheets, you can start working on the next batch while the first one bakes. You just don’t want the cookie dough to sit on the counter and warm up.
Bake. Place in the oven and bake for 15 minutes or until cracks form.
Cool. Remove the cookies from the oven and allow them to cool for 10 minutes on the baking sheet before transferring them to a wire rack to finish cooling for another 20 minutes. If you try to move the cookies before 10 minutes, they will be too soft and will break.