Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper or silicone mats for easy cleanup.
In the bowl of a stand mixer fitted with the paddle attachment, cream together the butter, brown sugar, and granulated sugar on medium speed until light and fluffy. Scrape down the sides of the bowl as needed.
Add the egg and vanilla extract, and continue mixing until the dough is smooth and well combined, about 1-2 minutes.
In a separate bowl, whisk together the flour, salt, baking soda, and cream of tartar until evenly combined. Gradually add the dry ingredients to the wet mixture, mixing on low speed until a thick dough forms.
Roll the dough into 1-inch balls (about 15 grams each) using your hands or a cookie scoop. Roll each ball in the cinnamon-sugar mixture until evenly coated, then place them on the prepared baking sheets. Use your thumb or the back of a small spoon to press a well into the center of each dough ball. If the dough cracks, gently press it back together.
Bake in the preheated oven for 7-8 minutes. The cookies should be slightly puffed with cracks forming on top. If the indentation has filled in during baking, simply press down gently again while the cookies are still warm.
While the cookies are cooling, prepare the eggnog ganache filling: In a microwave-safe bowl, combine the white chocolate, eggnog, and rum. Microwave in 15-second intervals, stirring between each, until the chocolate is fully melted and the mixture is smooth. Stir in the freshly grated nutmeg if you like. The ganache will be loose initially but will firm up as it cools.
Once the cookies are cooled, fill each thumbprint with about ½ teaspoon of the eggnog ganache. Top each cookie with a pinch of freshly grated nutmeg for an extra touch of holiday flavor.
Allow the cookies to sit at room temperature for a few hours to allow the ganache to set. For quicker results, you can refrigerate them for 15-20 minutes to firm up the filling.