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 ~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.
Place the milk, brown sugar and molasses in a medium mixing bowl and whisk until sugar is dissolved, either by hand or a hand-mixer on low speed.
Whisk in the pumpkin puree, cinnamon, ginger and nutmeg and stir until completely combined.
Slowly add in the heavy cream and vanilla, stirring until just mixed together. Place mixture in the fridge to cool for at least 1-2 hours.
Process ice cream mixture according to your manufactures instructions.
To assemble the sandwiches, scoop ice cream onto a cookie half and place another cookie half on top. Gently press down to seal, then place on a baking sheet or freezer-safe container. Continue with the rest of the cookies, layering them in a single layer so they don't stick together. Freeze until set, about 30 minutes.
Enjoy as is or wrap each sandwich in plastic wrap for longer storage in the freezer. Wrapped, they will keep roughly 3 months. Hope you enjoy these snickerdoodle and pumpkin ice cream sandwiches as much as I do!