1COOKIE: Mix together 1 cup butter with an on medium-high speed for 30 seconds. Add granulated sugar, the 3/4 teaspoon ginger, and the vanilla. Beat until combined, scraping bowl occasionally. Beat in as much of the flour as you can with the mixer. Mix in remaining flour and the 1 cup nuts by hand.
2Shape dough into 1-inch balls. Shape the moons by rolling each ball into a small log. Bend into a moon shape and place 1 inch apart on ungreased cookie sheets. Bake for 10 to 11 minutes at 350 or until set but not browned. Cool on a wire rack.
In a small saucepan heat butter until melted on medium heat. Keeping the butter at medium cook stirring constantly for about 7 minutes. the butter will bubble and turn golden. remove from heat and let cool slightly.
4in small bowl, combine brown butter, powder sugar and 1/4 teaspoon ginger. Stir in just enough milk to make a glaze that can spread (not too thick or thin). Once cookies are completely cooled, spread over cookies and sprinkle with the remaining nuts.