Place the powdered sugar in a medium bowl, and set aside.
Melt butter in a small saucepan over low-medium heat, stirring frequently, until it foams, the foam dissipates, it turns a deep golden brown, and develops brown flecks and a nutty fragrance, about 10 minutes or so. Toward the later stages, around the time it begins to foam, watch the brown butter very closely. It can go from 'brown' to 'burned' very quickly.
Remove from heat, cool 2-3 minutes, and *carefully* pour over powdered sugar.
Add milk and vanilla, and vigorously whisk the mixture until smooth. If the icing seems to seize or separate, you just need to reeeeeeally mix it to emulsify it. You can use an electric mixer if preferred. But don't give up thinking it's ruined, it just needs to be emulsified, so give it a serious mixing! Icing should be thin enough to spread, but not runny or drippy. Add additional milk or powdered sugar, as needed, to achieve desired consistency.
Spread approximately 1 tablespoon of icing onto each cookie. Allow to set. Enjoy!
Store cookies in an airtight container with parchment paper or foil between layers of cookies. At room temperature, cookies will keep for 4-5 days. They'll keep for about a week if stored in the fridge. They freeze beautifully if well-wrapped!