Whisk together flour, cream of tartar, baking soda, ½ teaspoon nutmeg, and salt.
Using a stand mixer fitted with the paddle attachment, beat butter, brown sugar, granulated sugar, and shortening together on medium speed until light and fluffy and pale in color, 5 minutes.
Scrape down sides of bowl, add egg, yolk, rum extract, and vanilla and beat until incorporated; scraping down sides of bowl as needed.
With mixer on low speed, add flour in 4 additions and mix just until combined; chill dough at least 45 minutes or up to 1 day.
Heat oven to 400°F (204ºC)with rack set in middle position; line two baking sheets with parchment paper.
Using a #24 cookie scoop, scoop dough then set on prepared sheets, 6 dough balls per sheet.
Bake cookies, one sheet at a time, until edges are lightly browned and centers are puffed but still look slightly raw, 10–12 minutes, rotating sheet halfway through baking.
Let cookies cool on sheet 5 minutes, then transfer to wire racks to cool completely. Repeat scooping and baking with remaining dough.
Sugar Dusting
If you prefer to finish cookies with sugar dusting: Whisk together confectioners’ sugar and ½ teaspoon nutmeg; transfer to a fine-mesh sieve and dust cookies with sugar mixture.
Eggnog Glaze
If you prefer to finish cookies with glaze: whisk together confectioners sugar, eggnog, nutmeg and salt. Drizzle over cookies and allow to dry 30 minutes before storing.
These cookies are best eaten within first 3 days of baking but last up to 1 week.