Preheat oven to 400 degrees F.
Place softened butter, 3/4 cup cane sugar, and brown sugar in the base of a stand mixer (or large bowl, using a hand-mixer). Cream the butter and sugar together until completely combined, for about 1 minute.
Beat in the eggs, one at a time.
In a separate large bowl, whisk the flour, cream of tarter, baking soda, salt and 1/2 teaspoon cinnamon. Slowly add the flour mixture into the sugar mixture until just combined.
Lightly spray a cookie sheet with cooking spray. In a small bowl, combine the remaining 1/4 cup sugar and 1 tablespoon cinnamon.
For sandwiches, you'll want the cookies to be uniform in size. I recommend using a tablespoon cookie scoop to form the balls. Scoop out cookie dough and roll in the sugar and cinnamon.
Place on the prepared cookie sheet and flatten slightly. Continue with the rest of the batter, placing the cookies about 3" apart.
Bake for 8-9 minutes until soft and tender. Ice cream sandwiches do better with a softer cookie, so watch your oven to make sure they don't become to crisp!
Remove and let cool completely on a wire rack.