Brown the Butter: In a medium saucepan, melt the unsalted butter over medium heat and cook until golden brown, about 5-7 minutes. Let it cool slightly.

Mix Wet Ingredients: In a large bowl, combine the cooled brown butter with brown sugar and granulated sugar, and beat until creamy. Add the egg and maple syrup, mixing until glossy.

Combine Dry Ingredients: In another bowl, whisk together the flour, baking soda, and salt. Gradually mix into the wet ingredients until just combined.

Chill the Dough: Cover the dough with plastic wrap and refrigerate for at least 30 minutes.

Preheat Oven & Bake: Preheat the oven to 350°F. Line a baking sheet and drop rounded tablespoons of dough onto it. Bake for 10-12 minutes until edges are golden.

Prepare Frosting: While cookies cool, whisk together browned butter, powdered sugar, and vanilla extract until smooth. Adjust consistency with milk if needed.

Frost Cookies: Spread the frosting generously on each cooled cookie and let set before serving.