Before baking the cookies, set the ice cream and pink lemonade concentrate out of the freezer to soften. In a large bowl, mix together the softened ice cream and lemonade concentrate. Return the ice cream mixture to its original container to refreeze overnight. You'll have extra lemonade-flavored ice cream after filling the sandwiches!
Preheat oven to 350 degrees.
In the bowl of your stand mixer, combine the butter and sugar. Beat on high for about 2 minutes until very light and fluffy. Add the egg and vanilla and beat for one minute more.
In a separate bowl, whisk together the flour, cream of tartar, baking soda and salt. Slowly add the flour mixture to the butter mixture and beat on medium in the stand mixer until well combined.
Scoop about 2 tablespoons of dough per cookie (I used an OXO cookie scoop) and dip the top of each cookie in sanding sugar. Place the cookies onto sheet pans lined with parchment paper or silicone baking mats. The recipe yields 20 cookies.
Bake cookies for 10 minutes, rotating pans halfway through baking time. The cookies will look a bit underdone. Allow to cool for at least 10 minutes on pans, then remove them to cooling racks to finish completely cooling.
Evenly scoop your vanilla ice cream and pink lemonade mixture in between pairs of cookies, yielding 10 ice cream sandwiches. Place sandwiches on a sheet pan to freeze for at least 2 hours before serving.