Combine the cream cheese and the sugar in a small bowl until smooth.
For the dough
Adjust an oven rack to the middle of the oven. Preheat the oven to 350F [180C]. Line three sheet pans with parchment paper.
In a medium bowl, combine the flour, baking soda, salt, and cream of tartar.
In the bowl of a stand mixer fitted with a paddle, beat the butter on medium speed until creamy, about 1 minute. Add 1 ¾ [350 g] of the granulated sugar and beat on medium speed until light and fluffy. 2 to 3 minutes. Add the egg and the yolk and beat on low until combined. Add the vanilla, red food coloring, and cocoa powder, and mix again on low until combined, scraping down the sides of the bowl. Add the flour mixture and beat on low speed until just combined.
Place the remaining ½ cup [100 g] of sugar in a medium bowl.
To assemble
Place 1 teaspoon of cream cheese in the top of a 1 ½ oz cookie scoop and use the back of spoon to spread it slightly over the inside of the scoop. Fill the scoop with sugar cookie dough (see note below if you don’t have a cookie scoop) and drop the dough into the bowl of sugar. Gently roll the dough around (it may be a bit sticky) to cover, and place on one of the prepared pans. Repeat with the remaining cream cheese and dough, putting 8 cookies on each sheet pan.
Bake one pan at a time, rotating halfway through baking. Bake until the sides are set, the centers are puffed and starting to form cracks, 12 to 13 minutes. Transfer the sheet pan to a wire rack and let the cookies cool for 5 to 10 minutes on the pan, then remove them and let them cool completely on the wire rack. Store cookies in an airtight container in the refrigerator for up to 2 days.
Notes
If you don’t have a cookie scoop, you can roll the dough into a ball, then dollop a scant teaspoon of cream cheese on the top of each cookie, using the back of a spoon to spread it over the top slightly. VARIATION: Green Mint Sugar CookiesAdd 1 teaspoon mint extract (more or less, depending on your preference) along with the pure vanilla extract. Replace the red food coloring with green. I used Wilton's Leaf Green gel-based food coloring, and was able to keep the cocoa powder in and still keep the cookie a vibrant green color. If you are using a liquid-based green food coloring, you will NEED TO OMIT THE COCOA, or your cookies will turn a brownish color. When using gel-based colorings you also don't need as much food coloring, so start out with just a little and then keep adding until the desired color is arrived at.