That egg carton looks like it makes a nice thick wad!
I put some thick folded postcard or card material from the junk mail pile over the rubber disk and adjust it just right to help keep the disk from getting cut and chewed up. The cutter will sometimes partially cut through a layer or 2 of the base paper depending on its thickness, but not all of the way if it's adjusted just right. But then the juice cartons that I'm cutting up are a thinner material that needs a little bit of support underneath them, especially when cutting 4 or 5 layers. The egg carton material probably doesn't need any support underneath. What a great idea!