Heat the oven to 350°F and line a baking sheet with parchment. The oven should be fully hot before baking.
Melt the butter in a small pan or microwave, then let it cool until warm but not hot. The butter should still be liquid.
Mix the melted butter with brown sugar and white sugar in a large bowl. Stir until smooth and glossy.
Add the eggs and vanilla. Stir until the mixture looks thick and fully blended.
Add flour, baking soda, and salt. Mix just until you no longer see dry flour.
Stir in the oats. The dough should look thick and slightly sticky.
Scoop dough into balls and place them a few inches apart on the baking sheet. Leave space so they can spread.
Bake for 8–11 minutes. The edges should look golden and the centers should still look soft.
Let the cookies cool on the pan for 5 minutes, then move them to a rack. They must cool completely before icing.
Whisk powdered sugar with milk and vanilla in a small bowl. The icing should be thin enough to drip slowly off a spoon.
Dip the top of each cookie into the icing or spoon it on. Place the cookies back on the rack.
Let the icing dry until it turns matte and crackles. This can take 30–60 minutes.