Brown the Butter
Melt butter in a light-colored saucepan over medium heat, stirring constantly as it foams and bubbles.
Watch for golden-brown specks to appear at the bottom and a nutty aroma to develop.
Remove from heat immediately and let cool for 20 minutes.
Prepare the Oats
Add old fashioned oats to a food processor and pulse until you have a mixture of fine crumbs and larger pieces. Do not pulverize into flour.
Make the Dough
Pour cooled brown butter into a large bowl. Add both sugars and whisk vigorously until well combined.
Add eggs and vanilla extract. Whisk until smooth and glossy.
Add molasses and stir thoroughly until completely mixed in.
In a separate bowl, whisk together flour, baking soda, cinnamon, nutmeg, and salt.
Add the dry ingredients and pulsed oats to the wet mixture. Switch to a spatula and fold gently until just combined. Do not overmix.
Chill the Dough
Scoop dough into equal-sized balls (about 2 ounces each) using a cookie scoop.
Place on a parchment-lined baking sheet, cover tightly with plastic wrap, and refrigerate for at least 3 hours or overnight.
Bake
Preheat oven to 350°F (175°C).
Bake one sheet at a time on the center rack for 9 to 10 minutes. Edges should be set while centers remain puffy and slightly underbaked.
Let cookies rest on the baking sheet for 5 to 10 minutes, then transfer to a wire rack to cool completely.
Ice the Cookies
Whisk together sifted powdered sugar, milk, and vanilla until thick like honey.
Dunk the top of each cooled cookie into the glaze, letting excess drip off before flipping upright.
Place on a wire rack and let set for 2 hours.