In place of the icing, try sprinkling these swirly doughnuts with a mixture of 2 tablespoons granulated sugar and 1/2 teaspoon pumpkin pie spice before baking.
As a guide, draw 3-inch circles with a finger in the fat used to prepare the pan. Fill a pastry bag with batter and fit with a star tip: Hold the bag as upright as possible and pipe out circles of batter.
As the doughnuts cool, prepare the icing. Hold the spoon over each doughnut and let the icing fall from the spoon; it will glide through the channels created by the star tip and cover the surface.