Preheat oven to 350 degrees F (175 degrees C). Lightly grease baking sheets.
In a large bowl, mix together the cake mix, butter, vanilla extract, eggs, chocolate chips, and butterscotch chips. Roll dough into 2-inch balls and place them onto the prepared baking sheets. Bake for 10 minutes in the preheated oven. Do not overbake; the cookies should be set, but still soft. Let cool on the baking sheet for a few minutes before removing to finish cooling on wire racks.
Using an ice cream scoop, mold the softened ice cream into a ball about the size of the cookies. Place scoop of ice cream between two cookies and gently press together. If you like a smooth edge, run the back of a spoon around the edge of the ice cream sandwich. Working quickly, fill all of the cookies, lay out on a baking sheet, and place in the freezer to harden. Serve when the ice cream has frozen solid. Sandwich cookies can be stored in a freezer bag for up to two weeks.