Add granulated sugar, vanilla extract and egg to the butter and mix well.
Add cake flour in several batches, stirring well after each addition, then knead by hand.
Divide the kneaded cookie dough into 5 to 6 pieces.
Chill the dough for at least an hour in the refrigerator.
Roll the dough out about 5mm thick, and cut out with a cookie cutter.
Bake in a 340°F/170°C oven for about 10 minutes.
For the icing:
Mix the dried egg white powder and water and let sit for 30 minutes.
In the meantime, sift the powdered sugar.
Add the sifted powdered sugar to the egg white liquid, and mix until shiny.
Put the icing in a piping bag, and draw an outline around the edges of each cookie.
Wait until the outlines have totally dried.
Add water to the remaining icing to soften.
Pipe in the softened icing inside the outlines to fill them in.
Wait until the icing has dried thoroughly.
Mix the black food coloring in the icing.
Add a tiny bit of water to the icing, so that's it's a consistency that's easy to draw with.
Draw the faces on, with the piping bag held perpendicular to the cookie surface.
Apply the decorations:
Stick on the decorations before the icing dries.
Here's how the cookies look.
A close up view.
Story Behind this Recipe
I adapted the cookie recipe from one in a cookbook called "DIA Collection's Frosted Cookie Decoration Recipe." I think that this only needs black icing instead of several colors, which simplifies the recipe.
Roll the cookie dough out thinly. That'll make the cookies easier to ice and better looking. Even a 1mm icing tip may be too wide. It's easier to cut a tip off a plastic icing or piping bag so you can draw a thin line. You can use the cheap ones they sell at a 100 yen shop (Japanese equivalent of a dollar store). Make the icing outlines really really thin.