Preheat oven to 350 degrees F. Lightly grease a cookie sheet; set aside. In a large bowl beat the 1 cup butter with an electric mixer on medium to high speed for 30 seconds. Add brown sugar, baking soda, and salt. Beat until combined, scraping bowl occasionally. Add maple syrup, egg, and vanilla; beat until combined. Beat in as much of the flour as you can with the mixer. Using a wooden spoon, stir in any remaining flour.
In a medium bowl whisk together evaporated milk, melted butter, and maple flavoring until combined. Gradually whisk in powdered sugar until mixture is icing consistency. Spread tops of cooled cookies with icing.