Transfer to a cookie sheet lined with parchment paper or a silicone mat.
Bake for 8-10 minutes or until the giant cookie is pale golden and the edges have just set. The cookie will still be really soft.
Remove from the oven and set on the counter for 3 minutes.
Take cola ice cream out of the freezer to warm up slightly. You'll want it soft but not soupy.
Use the cookie cutter portion of the ice cream sandwich maker to to cut out star cookies.
Place the cut-out cookies on a cooling rack to finish cooling. Eat the scraps.
Place a cooled cookie in the bottom of the ice cream sandwich maker, top with a scoop of ice cream and another cookie, and use the cookie press to tightly pack the whole thing.
Remove the ice cream sandwich from the mold, tightly wrap in plastic wrap, and freeze until ready to eat.