Beat butter and shortening in a large mixing bowl with an electric mixer on medium to high speed for 30 seconds. Add granulated sugar, brown sugar, baking soda, and salt; beat until combined, scraping sides of bowl occasionally. Beat in egg, milk, and vanilla until combined. Beat in as much flour as you can with mixer. Stir in remaining flour with a wooden spoon.
Divide dough in half. Into one half, stir red food coloring until well mixed and desired color. Divide red dough and plain dough into thirds. Line the bottom and two sides of an 8x4x2-inch loaf pan with waxed paper. Place one portion of red dough between two pieces of waxed paper. Roll or pat dough into a 7x2-inch rectangle. Remove waxed paper. Place in bottom of prepared loaf pan. Gently pat dough to edges of pan. Repeat rolling and patting with remaining portions of dough, alternating red and plain doughs. Cover loaf pan with plastic wrap. Freeze at least 4 hours or overnight.
To bake, grasp the waxed paper to lift the dough out of the loaf pan. With a sharp knife, slice the dough crosswise into three equal parts. Then, starting at the short end of each part, cut crosswise into 1/4-inch-thick slices. Place slices 1 inch apart on an ungreased cookie sheet.
Bake in a 375 degree F oven for 8 to 10 minutes or until edges are golden. Transfer cookies to a wire rack and cool. Spread a 1-inch square of white frosting on the upper left corner of each flag. Make a star in the center of each frosting square with blue decorator gel. Makes about 36 cookies.