Heat oven to 350°F. Line two baking sheets with parchment paper.
Beat butter and cream cheese until smooth and creamy, about 1–2 minutes.
Add granulated and brown sugars. Cream until light and fluffy.
Mix in lemon zest, lemon juice, and vanilla until combined.
Whisk flour, cornstarch, baking powder, baking soda, and salt in a bowl.
Add dry ingredients to the wet mixture in two additions. Mix just until no dry streaks remain.
Fold in blueberries gently. Chill dough 15–20 minutes to firm slightly.
Scoop about 1½ tablespoons of dough per cookie onto sheets, spacing 2 inches apart.
Bake 10–12 minutes, until edges set and tops look matte. Cool 5 minutes, then transfer to a rack.
Whisk confectioners' sugar with lemon juice and a splash of milk to a drizzle. Spoon over cooled cookies and let set.