Melt the butter in a light-colored saucepan over medium heat, stirring, until it foams, then turns golden with brown flecks and a nutty aroma, about 5 minutes. Scrape it, flecks and all, into a bowl to cool. Brown the dough butter and icing butter separately.

Press the pumpkin between paper towels to blot out excess water.

Whisk together the flour, oats, pumpkin pie spice, cinnamon, baking soda, and salt.

Whisk the cooled brown butter with the brown sugar and granulated sugar until smooth. Stir in the blotted pumpkin, egg, egg yolk, and vanilla.

Stir the dry ingredients into the wet just until combined, then let the dough rest 10 minutes.

Preheat the oven to 350°F and line two baking sheets with parchment. Scoop rounded tablespoons about 2 inches apart.

Bake 12 to 14 minutes, until the edges are set but the centers look soft. Cool on the pan 5 minutes, then transfer to a rack to cool completely.

Whisk the second batch of browned butter with the sifted confectioners' sugar.

Whisk in the milk a tablespoon at a time and the vanilla until smooth and pourable. Drizzle over the cooled cookies and let set.