Cream the butter and sugar in a large mixing bowl until smooth using an electric mixer. Add the eggs and mix until combined.
Sift all the remaining dry ingredients in a separate large mixing bowl and mix well. With the mixer on low, slowly add the dry ingredients to the butter mixture until a dough forms. Remove dough from the bowl and wrap in plastic wrap. Refrigerate until firm, 2 to 4 hours.
Preheat the oven to 350 degrees F and grease cookie sheets.
Remove dough from refrigerator and allow dough to sit at room temperature for about 15 minutes, until pliable. Divide dough in half and put on a floured work surface. Using a rolling pin, roll the dough out until it's about 1/8-inch thick. While rolling, flour the work surface and rolling pin, as needed.
Cut out desired cookie shapes with cookie cutters. Transfer cookies to a greased cookie sheet, spacing them 1-inch apart.
Lightly brush cookies with milk and sprinkle with sugar granules, as desired.
Bake until firm and edges begin to darken, 10 to 14 minutes. Remove the cookies from the oven and allow to cool on the cookie sheet for a minute, then transfer to wire racks to cool completely.
You can find star anise in the ethnic aisle in your local grocery store. To get the best flavor from this spice, you will need to grind the whole star anise pod. For the neatest edges, dip your cookie cutters in flour before cutting out each cookie.
A viewer or guest of the show, who may not be a professional cook, provided this recipe. It has not been tested for home use.