Preheat oven to 350F. Line two medium sheet pans with parchment paper and set aside.
Place butter and 1 cup sugar in the bowl of a large stand mixer, fitted with a paddle attachment OR in a large mixing bowl with a wooden spoon. Mix on medium speed or by hand until creamy, about 2 minutes. Add egg and vanilla, beating well until smooth.
In a separate medium bowl, combine flour, cream of tartar, baking soda, and salt. Fold into butter mixture until dough is just combined.
In a small bowl, combine remaining 1/4 cup sugar and cinnamon.
Use a medium cookie scoop to scoop dough out into 1-in balls. Roll balls in cinnamon sugar until evenly coated and place 2 inches apart on prepared sheet pans.
Bake cookies at 350F 10-12 minutes until edges of cookies are golden brown. Allow cookies to cool on sheet pans 10 minutes before transferring to a wire cooling rack to cool completely.
Scoop ice cream into 1/4 cup portions and sandwich between 2 snickerdoodles. Place filled cookies on a plate and immediately transfer to freezer. Freeze sandwiches 2 hours until firm. Enjoy!