Mix together the brown sugar, 1 teaspoon maple extract, vanilla, 3/4 cup butter and egg. Stir in the flour mixture. The dough will be very sticky.
Drop onto greased cookie sheet. Bake for 10-12 minutes at 350 degrees F (175 degrees C). Let rest for one minute before removing from cookie sheet. Frost when cool.
To Make Frosting: Combine 2 cups confectioners' sugar, 2 tablespoons butter, 2 tablespoons milk, and 1 teaspoon maple extract. Spread over cooled cookies.
