Brown the Butter: Cut butter into pieces and melt in a light-colored saucepan over medium heat. Continue cooking, stirring frequently, until milk solids turn golden brown and butter smells nutty, 5 to 7 minutes. Immediately pour into a heatproof bowl and cool for 20 minutes.

Pulse Oats: Add oats to food processor and pulse 10 to 15 times until you have a mixture of fine crumbs and larger oat pieces. Set aside.

Mix Dry Ingredients: In a bowl, whisk together flour, baking soda, cinnamon, nutmeg, and salt. Set aside.

Mix Wet Ingredients: Add sugar to cooled brown butter and whisk vigorously for 1 minute until thick and paste-like. Add eggs and vanilla, whisking until smooth. Add molasses and stir until no streaks remain.

Combine: Add flour mixture to wet ingredients and fold gently with spatula just until combined. Add pulsed oats and fold until evenly distributed. Do not overmix.

Chill Dough: Scoop dough into equal-sized balls (about 2 tablespoons each) and place on parchment-lined baking sheet. Cover with plastic wrap and refrigerate for at least 3 hours or overnight.

Bake: Preheat oven to 350°F (175°C). Bake chilled cookies one sheet at a time on center rack for 9 to 10 minutes. Edges should be set and centers slightly underdone. Cool on baking sheet for 5 to 10 minutes, then transfer to wire rack to cool completely.

Make Glaze: Whisk together powdered sugar, 2 tablespoons milk, and vanilla extract until smooth. Glaze should be thick enough to slowly drip from whisk. Add more milk or sugar to adjust consistency.

Glaze Cookies: Pour glaze into shallow bowl. Dunk top of each completely cooled cookie into glaze, let excess drip off, and place on wire rack right-side up. Let sit for 2 hours until glaze fully sets.