Preheat the oven to 350 (F). Add butter and brown sugar to a bowl. Use a hand mixer or stand mixer and mix for 3-4 minutes. Add eggs and vanilla and mix until just combined.
Add flour, cornstarch, baking soda, and salt. Mix until just combined. The dough should be very soft.
Add the extra brown sugar to a shallow bowl or plate. For extra large cookies, scoop out 1/2 cup of dough and roll it into a ball. Transfer the dough ball to the brown sugar and roll it around until the dough ball is covered in brown sugar. Transfer the dough ball to a lined baking sheet and repeat with the remaining dough (you can make the cookies smaller if you want, just take some time off of the cook time!).
If you're making large cookies, add 4-6 cookies to a large baking sheet. Flatten the dough balls slightly so they are more like large cylinders. Bake in the oven for 9-11 minutes, until the cookies are just starting to set.
While the cookies cool, make the glaze. Add the cream cheese, powdered sugar, maple syrup, and vanilla to a bowl. Whisk until the mixture is smooth. Once the cookies are cool, drizzle the glaze onto each cookie and enjoy!