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1. Heat butter in a small saucepan over medium heat until melted. Reduce heat to medium-low to low, and heat until butter turns the color of light brown sugar (about 12 minutes). Remove from heat. Pour into a large bowl; chill until butter just resolidifies (about 1-1/2 hours) or freeze about 45 minutes.
2. Preheat oven to 375 degrees F. Set aside 4 teaspoons of the browned butter for the filling. Beat remaining 2/3 cup of the butter with an electric mixer on medium to high speed for 30 seconds. Add granulated sugar, baking powder, and salt. Beat until combined, scraping bowl occasionally. Beat in egg, milk, and 1 teaspoon of the vanilla. Beat in as much of the flour as you can with the mixer. Stir in any remaining flour. Divide dough in half.
3. On a lightly floured surface, roll half of the dough at a time to 1/8 inch thick. Cut dough into stars using a 2-1/2-inch cookie cutter. If desired, using a small cutter, cut desired shapes from centers of half of the cookies. Place cutouts 1 inch apart on ungreased cookie sheets.* Reroll scraps as necessary.
4. Bake in the preheated oven for 7 to 8 minutes or until edges are firm. Transfer to wire racks and let cool. Repeat with remaining dough half.
5. For filling, in a medium bowl beat the reserved browned butter with an electric mixer for 30 seconds. Beat in remaining 1 teaspoon vanilla and the powdered sugar until combined. If necessary, beat in milk, 1 teaspoon at a time, to make an icing of soft spreading consistency. Tint icing with green and/or red food coloring.
6. Spread icing over the flat sides (bottoms) of the cookies with no cutout centers (or half the cookies if none have cutouts). Top with remaining cookies, flat sides down, offsetting points. Makes about 18 sandwich cookies.