In a saucepan, melt the butter over medium heat until it turns golden brown and has a nutty aroma, about 5-7 minutes. Remove from heat and let cool slightly.

In a mixing bowl, combine the brown butter, brown sugar, and granulated sugar. Mix until well combined.

Add the egg and vanilla extract, mixing until smooth.

In another bowl, whisk together the flour, baking soda, and salt. Gradually add to the wet mixture, stirring until just combined.

Fold in the oats and chocolate chips (if using).

Preheat the oven to 350°F (175°C). Line a baking sheet with parchment paper.

Scoop tablespoon-sized portions of dough onto the prepared baking sheet, spacing them about 2 inches apart.

Bake for 10-12 minutes or until the edges are lightly golden. Allow to cool on the baking sheet for 5 minutes before transferring to a wire rack.

In a small bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth. Adjust the milk for desired consistency.

Drizzle the icing over the cooled cookies.