Brown the butter: Cut butter into tablespoon-sized pieces and place in a light-colored saucepan over medium heat. Stir constantly as butter melts, foams, and develops golden-brown specks, about 5-7 minutes. Once amber-colored and fragrant, immediately pour into a heatproof bowl and cool for 20 minutes.

Prepare oats: Pulse old-fashioned oats in a food processor until you have a mixture of fine crumbs and larger pieces. Set aside.

Mix wet ingredients: In a large bowl, whisk cooled brown butter with both sugars until combined. Add eggs one at a time, whisking well after each. Stir in molasses and vanilla extract.

Combine dry ingredients: In a separate bowl, whisk together flour, baking soda, cinnamon, nutmeg, and salt. Add pulsed oats and stir to combine.

Form dough: Add dry ingredients to wet ingredients and fold gently with a spatula until just combined. Do not overmix. Cover bowl with plastic wrap and refrigerate for at least 3 hours or overnight.

Shape and chill: Scoop chilled dough into 1.5-ounce balls and place on parchment-lined baking sheets. Cover with plastic wrap and refrigerate while preheating oven to 350°F.

Bake: Bake one sheet at a time on center rack for 9-10 minutes, until edges are set and centers are puffy and slightly underbaked. Cool on baking sheet for 5-10 minutes, then transfer to wire rack to cool completely.

Make glaze: Whisk together powdered sugar, milk, and vanilla until thick but pourable. Add more milk if needed to reach proper consistency.

Glaze cookies: Pour icing into a shallow bowl. Dunk top of each completely cooled cookie into glaze, allowing excess to drip off. Place on wire rack and let set for 2 hours before serving or storing.