Preheat oven to 325 degrees F (165 degrees C). Line 2 baking sheet with silicone baking mats.

Beat cake mix, eggs, 1/2 cup flour, 3 tablespoons flour, and soybean oil together in a stand mixer until stiff.

Divide and roll dough into 1-inch balls and arrange on prepared baking sheets and flatten to about 1/2-inch thick.

Bake in preheated oven until somewhat firm in the center, 12 to 14 minutes. Remove from oven and let cookies sit undisturbed until set, about 5 minutes more.

Spread a sheet of aluminum foil onto a countertop. Transfer cookies to foil to cool completely, at least 20 minutes.

Beat confectioners' sugar, butter, shortening, half-and-half, and vanilla extract together with a hand mixer until smooth; spoon into a pastry bag and pipe onto cooled cookies or just spread onto cookies using a spatula.