Prepare the Dough : In a large bowl, cream together butter, light brown sugar, and white granulated sugar until light and fluffy. Add the egg and vanilla extract, mixing until fully incorporated. In a separate bowl, whisk together all-purpose flour, salt, baking soda, and cream of tartar. Gradually add the dry ingredients to the wet mixture, mixing until a soft dough forms.
Chill the Dough : Cover the dough with plastic wrap and refrigerate for at least 30 minutes.
Roll and Shape : Preheat the oven to 350°F (175°C). Line a baking sheet with parchment paper. Roll the dough into 1-inch balls and coat them in the cinnamon-sugar mixture. Place the balls on the prepared baking sheet, spacing them about 2 inches apart. Use your thumb or the back of a small spoon to create a small indentation in the center of each cookie.
Bake : Bake the cookies for 8–10 minutes, or until the edges are set but the centers remain soft. Remove the cookies from the oven and gently press down on the indentations again, if needed. Let them cool on the baking sheet for 5 minutes before transferring to a wire rack.
Prepare the Eggnog Filling : While the cookies are cooling, melt the white chocolate in a heatproof bowl over a pot of simmering water or in the microwave in 20-second intervals, stirring frequently. Stir in the eggnog, dark rum (or rum extract), and freshly grated nutmeg until smooth.
Fill the Cookies : Spoon the eggnog filling into the indentations of each cookie. Allow the filling to set for 15–20 minutes at room temperature or refrigerate briefly to speed up the process.
Serve and Enjoy : Enjoy these cookies fresh or store them in an airtight container at room temperature for up to 3 days.