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Preheat oven to 350°F. Lightly grease a cookie sheet; set aside. In a large bowl beat 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. Stir in any remaining flour.
Drop dough by rounded teaspoons 2 inches apart onto prepared cookie sheet; flatten slightly. Bake in the preheated oven for 8 to 10 minutes or until tops are set. Transfer cookies to a wire rack; let cool.
In a medium bowl, whisk together evaporated milk, melted butter, and maple flavoring until combined. Gradually stir in powdered sugar until mixture is icing consistency. Spread tops of cooled cookies with icing.