Preheat oven to 350F. Line two large baking sheets with parchment paper.
In a medium bowl, whisk together flour, baking soda, and salt until combined.
In a large mixing bowl or the bowl of a stand mixer fitted with the paddle attachment, beat the butter and sugar at medium speed until light and fluffy, 2 to 3 minutes. Stop and scrape down the sides of the bowl as needed. Beat in the egg and vanilla until well combined.
Add the flour mixture and beat on low speed until combined, stopping to scrape down the bowl once during mixing. Turn the dough out onto a lightly floured surface and shape it into a disk. Wrap the dough in plastic wrap and chill for 1 hour or until firm.
On a lightly floured surface, roll out the dough to about 1/4-inch thickness. Dip a 2½-inch cookie cutter in flour and cut out 12 rounds. (Reroll the scraps if necessary.) Place the rounds 3 inches apart onto baking sheets.
Bake, one sheet at a time, for 8 to 10 minutes until the centers appear dry and edges are lightly golden brown. Let cool on the pan for 5 minutes, then transfer to a wire rack to cool completely. Store cakes in an airtight container at room temperature for up to 1 week.
Notes
Bake 6 cakes at a time. The cakes will spread as they bake, so it’s best to place just 6 cakes per baking sheet so they have plenty of room to expand without touching. Select a large enough baking sheet to allow 3 inches between cookies.
Use nutmeg to add color and flavor. A bit of nutmeg will add color and an eggnog-like flavor. Sprinkle the tops of the cakes with nutmeg before baking, or add ¼ teaspoon of grated nutmeg to the dry ingredients. Freshly grated nutmeg has a warm, nutty, and festive flavor.
Cut into round, scalloped, or mini cakes. Use a 2½-inch cookie cutter or biscuit cutter with round or scalloped edges, or try a smaller cutter to make mini cakes. Be sure to dip the cutter in flour as needed to prevent sticking. Smaller cakes will bake more quickly, so watch for their edges to become brown after 6-8 minutes.
Re-roll the scraps of dough. After cutting the dough into about 12 cakes, you can re-roll the scraps of dough and cut additional cakes, if desired. Gather the scraps and press them into a ball using the palms of your hands. Re-roll the dough, cut additional cakes, and repeat as necessary.