In an electric stand mixer, cream the butter and brown sugar until light and fluffy.
Mix in the eggs and maple extract.
Finish with the dry ingredients and mix until just combined.
Using a 2-ounce cookie scoop, place cookies on parchment-lined baking sheets. Bake in preheated oven until puffed and just set in the middle, 10-12 minutes.
Remove and cool on pan for 5 minutes. Finish cooling on rack.
While baking and cooling, make brown butter frosting.
For the frosting
Melt butter over medium heat. Continue to cook until the foaming subsides and butter turns a brown nutty color, 6-8 minutes.
Remove from heat and whisk in powdered sugar and buttermilk. Start with less buttermilk and add more as needed.