In an electric mixer, use the paddle attachment to cream the butter for about 1 minute on medium speed. With the mixer still at medium speed, add the sugar and beat until the mixture is fluffy and light yellow. Then add in the egg, egg yolk, and vanilla, scraping the bowl as needed to ensure everything is mixed in thoroughly.
In a separate medium bowl, whisk the flour, baking soda, cream of tartar, cornstarch, and salt together. Add this to the wet ingredients in three portions with the mixer on low speed. Let the mixer go until the dough comes together. Now, the most tantalizing part. With the mixer on low, add your sprinkles. Watch the rainbow distribute. It is epic.
Form the cookie dough in 1½ tablespoon balls and chill on a cookie sheet lined with parchment paper, covered. 9 cookies should fit comfortably on one sheet. We rolled some for baking, and the rest went into a gallon ziploc bag in a single layer and into the freezer for another day! You should be able to make about 24-30 cookies using this amount of dough.
After the chilling is complete, remove the cookie dough from the refrigerator and let sit on the counter for 15 minutes. In the meantime, preheat your oven to 325 degrees F. Bake the cookies (one cookie sheet at a time) for 11-12 minutes, or until the edges are just slightly browned. They should be quite soft in the centers fresh out of the oven (if you’re baking them straight from the freezer, tack on an extra 1-2 minutes of baking time). Let the cookies cool on the sheet for 3 minutes, then transfer to a wire rack to cool completely.
Grab your favorite vanilla ice cream (or any other flavor you care to enhance your cookies with). Let it soften out on the counter for about 5-10 minutes, depending on how hot/cold your kitchen is.
When the ice cream is softened, scoop about a half cup onto one cookie and gently use the second cookie to form the sandwich. You might need to use a spatula to smooth out the edges. You can make these one at a time and devour immediately, but the better way is to make a bunch, wrap them in saran, and toss back into the freezer for at least 30 minutes to let them firm back up. You can also make them ALL and freeze them so you can have a summery treat whenever your heart desires!