Beat the softened butter and granulated sugar for 2–3 minutes, until light and creamy. Add the egg and vanilla extract, then mix until fully combined.

Whisk the flour, baking powder, and salt in a separate bowl. Add the dry mixture gradually to the butter mixture, mixing only until a soft dough forms.

Divide the dough into two portions, flatten each portion into a disk, wrap, and chill for at least 30 minutes. Chilling makes the dough easier to roll and helps the cookies keep their donut shape.

Heat the oven to 350°F (175°C) and line baking sheets with parchment paper. Roll one dough disk on a lightly floured surface to approximately ¼ inch (6 mm) thick.

Cut donut shapes with a donut cookie cutter, or use a plain round cutter and remove a smaller circle from the center. Transfer the shapes to the prepared baking sheets, spacing them about 1 inch apart.

Bake for 8–11 minutes, until the edges are set and only very lightly golden. Cool the cookies on the baking sheet for 5 minutes, then transfer them to a rack to cool completely.

Whisk the confectioners’ sugar, milk or water, vanilla, and optional corn syrup into a thick but spreadable icing. Divide it into bowls and tint each portion with gel food coloring.

Dip or spread icing over the cooled cookies, then add Christmas sprinkles or nonpareils while the icing is wet. Let the decorated cookies set for about 30–60 minutes before stacking or packaging.