Whisk powdered sugar, maple syrup, and milk into brown butter until thick but pourable; dip/spread over cooled cookies and let set.
Beat softened butter with brown sugar and granulated sugar until fluffy (2–3 min).
Mix in egg, maple syrup, and vanilla until combined.
Mix dry ingredients into wet on low just until no flour streaks remain.
Cover and refrigerate at least 1 hour (prevents spreading).
Scoop onto lined sheet, bake at 350°F for 10–12 min until edges set and centers look slightly soft. Cool 5 min, then move to rack.
Melt butter in a light pan, cook until golden with nutty aroma; remove and cool 10 min.
Whisk powdered sugar, maple syrup, and milk into brown butter until thick but pourable; dip/spread over cooled cookies and let set.