In a large bowl, whisk together the cake flour, all-purpose flour, baking powder, and cream of tartar until the mixture smells faintly of starch and spring. Set aside.

In another bowl, beat together the sugar, salted butter, and butter-flavored shortening until the mixture is pale and airy — it should hold ribboned traces on the beater.

Add the vanilla extract and the whole egg, mixing until combined; then gently fold in the egg white. The batter will look glossy and forgiving.

Gradually incorporate the dry ingredients into the wet, mixing just until the dough comes together. You should feel it becoming smooth and cool beneath your hands.

Cover the dough and refrigerate overnight.

When ready to bake, preheat the oven to 375°F and line baking sheets with parchment paper.

On a lightly floured surface, roll the dough to 1/2 inch thick. Cut out shapes with your chosen cutters, dust each lightly with the 1/3 cup of sugar, and place them on the prepared baking sheets.

Bake for 10 to 12 minutes, until the edges are firm and just beginning to color. The center should look set but not overbrowned.

Transfer to a wire rack to cool completely.

Meanwhile, warm the frostings in short bursts in the microwave until pourable but not boiling.

Dip the tops of the cooled cookies into the warm frosting, allowing excess to drip back into the bowl, then place each cookie on parchment.

Sprinkle with nonpareils and other decorations while the glaze is still wet.

Allow the cookies to dry for about 1 hour before stacking or packaging.