Preheat oven to 350 degrees (325 if you are using a dark metal pan). Line 13x9 baking pan with parchment paper. Lightly spray with nonstick cooking spray. Set aside.
Place melted butter into a medium bowl. Add 1/2 cup of white sugar and 1/3 cup of brown sugar to bowl and whisk together until smooth and when the butter has cooled down. Add vanilla extract and egg and whisk together until smooth. Add flour and cream of tartar and fold with a spatula or wooden spoon just until combined. Do not overmix!
Spoon batter into prepared pan and smooth into an even layer. The layer will be pretty thin so an offset spatula really help push it into all the corners.
Mix remaining 1/4 cup of sugar and ground cinnamon in a small bowl. Sprinkle the sugar mixture over the batter so that the whole top is covered in an even layer of cinnamon sugar. Use all of it!
Bake in preheated oven until set and edges just begin to turn brown, about 15 minutes. Make sure the center is cooked all the way by checking with a toothpick. Remove from oven and let cool completely and then freeze for 30 minutes.
Let vanilla ice cream soften at room temperature for about 10 minutes. Remove chilled snickerdoodle layer from freezer and cut in half lengthwise to make two halves. Flip one half upside down so the cinnamon sugar layer is facing down. Top with scoops of vanilla ice cream and spread into an even layer. Top with other layer of snickerdoodle cookie, cinnamon sugar facing up. Press down gently and place back into 13x9 pan and into the freezer to chill for 6-8 hours, or until firm.
Once ice cream has firmed to a solid/semi-solid state, it's time to cut the sandwiches. You can cut into three strips and then into 3 large sandwiches per strip or 6 small sandwiches per strip. Wrap each one in plastic wrap and store in freezer until ready to consume. S