There's one type of cookie that you probably haven't used to make an ice cream sandwich, but should, as its texture holds up to ice cream well: the drop cookie.
While many cookies are rolled out for a uniform look and placed on the baking sheet in an orderly fashion, drop cookies are dropped onto the baking sheet, making them thicker.
This textural difference makes drop cookies great for ice cream sandwiches because a thinner cookie type may crumble or break when it has ice cream scooped onto it.
Because the cookies "drop" after you make the batter, you can use almost any kind of cookie, so the possibilities are endless for cookie flavors in your ice cream sandwiches.
Try making oatmeal cookies with raisins or chocolate chips for a more eclectic mouthfeel or score peanut butter cookies after you "drop" them for unmatched texture contrasts.