Math theorem: the most misshapen ham sandwich can always be cut into two perfect halves

There is a meta-problem here. At what point does a sandwich cease to be a sandwich anymore, so that the act of cutting it in half becomes meaningless? If the two pieces of bread are so misaligned as to only have less than 50% overlapping and some portion of the meat itself is off the bread entirely, it’s no longer a sandwich. It’s like adding barbecue chicken to pizza. At that point, it’s no longer pizza.