Preheat the oven to 375°F, making sure the rack is in the center. Line two 11x17-inch cookie sheet with parchment paper.
Place the butter and sugar in the bowl of a stand mixer fitted with the paddle attachment. Cream them together on medium speed until the mixture has lightened in color and is fluffy, about 2 minutes.
Add the applesauce and vanilla along with a spoonful of the flour and mix until combined.
Sift the cream of tartar, cinnamon, baking soda, salt, and the rest of the flour into the mixture and mix on medium-low speed until the dough is evenly mixed.
Stir together the additional sugar and cinnamon in the small bowl. This is for rolling the dough balls before baking them.
Roll the dough into 1 tablespoon-sized balls, then roll the balls in the cinnamon sugar. Place 12 balls per tray, spacing them equally apart.
Bake the cookies at 375°F for 8 minutes per tray. The edges of the cookies will be set, but they will look soft, puffy, and underbaked. This is how they should look when they come out, as they will flatten and firm up considerably as they cool.
Let the cookies cool for 1-2 minutes on the tray, then transfer them to a wire to cool. Enjoy while warm or at room temperature. Repeat steps 6-8 with any remaining cookie dough.