In a medium bowl, whisk together the flour, cream of tartar, baking soda, and ½ teaspoon freshly grated nutmeg. Set aside. In the bowl of a stand mixer with the paddle attachment, cream together the butter, 1 ¾ cups sugar, and the salt on medium-high until very smooth and fluffy, 3 to 4 minutes.
Scrape down the sides of the bowl, return the mixer to medium speed, and add the egg and egg yolks, incorporating each before adding the next. Add the rum extract. Scrape down the sides of the bowl again and continue beating for another minute or two.
Add the flour mixture and beat on low speed until combined. (You may want to start by “pulsing” the flour, so that it doesn’t go everywhere.) Set the dough aside to hydrate for about 10 minutes while you prepare to bake.
Preheat the oven to 350 degrees and line 2 baking sheets with parchment paper. Make the nutmeg-sugar mixture to roll the cookies in by combining the remaining ¼ cup sugar and 2 teaspoons nutmeg in a small bowl.
Roll the dough into golf ball-sized balls, the roll each dough ball in the nutmeg-sugar. Transfer the balls to the baking sheets, about 2 inches apart. Barely flatten the balls to about ¾-inch thick with the back of a drinking glass. If using, gently press the dragées into the tops of the cookies. Bake 10 to 13 minutes, rotating pans and switching racks halfway through, until slightly puffed and just set.
Let the cookies cool slightly on the baking sheet before transferring them to a wire rack to cool completely. These cookies are even better the day after they’re made.