01 - Preheat oven to 350°F (175°C). Line two baking sheets with parchment paper.
02 - In a large bowl, cream together the softened butter and granulated sugar until light and fluffy. Beat in the large egg and vanilla extract until well combined. Gradually add the all-purpose flour, cream of tartar, baking soda, and salt, mixing until just incorporated to form a cohesive dough.
03 - Roll rounded tablespoons of dough into balls and then roll each ball in the cinnamon sugar mixture. Place the coated balls about 2 inches apart on the prepared baking sheets. Bake for 9-11 minutes, or until the edges are lightly golden brown. Allow the cookies to cool on the baking sheets for a few minutes before transferring them to a wire rack to cool completely.
04 - In a medium bowl, beat the softened butter until creamy. Gradually add the powdered sugar and ground nutmeg, mixing until combined. Add the eggnog, 1 tablespoon at a time, beating until the frosting reaches a smooth, spreadable consistency. Add the pinch of salt and mix well.
05 - Once the cookies are completely cool, spread a dollop of eggnog buttercream onto the flat side of one cookie and top with another cookie to create a sandwich. Repeat with the remaining cookies. Chill the assembled sandwich cookies in the refrigerator for at least 10 minutes to allow the buttercream to set.