Heat oven to 350°F.
Combine 1 cup butter, sugar, 1 teaspoon ginger and rum extract in bowl. Beat at medium speed, scraping bowl often, until well mixed. Add all remaining cookie ingredients; beat at low speed until well mixed.
Shape dough into 1-inch balls; form into crescents. Place 1 inch apart onto ungreased cookie sheets. Bake 9-11 minutes or until set. Cool completely.
Melt 1/4 cup butter in heavy saucepan over medium heat. Cook, stirring constantly and watching closely, 4-6 minutes or until butter just begins to turn golden brown. Immediately remove from heat. Cool 1 minute on cookie sheet; remove to cooling rack placed over waxed paper.
Combine browned butter, powdered sugar, 1/2 teaspoon ginger and enough half & half for desired glazing consistency in bowl. Drizzle cooled cookies with glaze. Immediately sprinkle with finely chopped macadamia nuts.
To shape into a crescent, roll the ball of dough between your palms; form a short log with tapered ends and curve slightly into a crescent shape.