The ham sandwich theorem is also sometimes referred to as the “ham and cheese sandwich theorem”, again referring to the special case when n = 3 and the three objects are
1. a chunk of ham,
2. a slice of cheese, and
3. two slices of bread (treated as a single disconnected object).
The theorem then states that it is possible to slice the ham and cheese sandwich in half such that each half contains the same amount of bread, cheese, and ham. It is possible to treat the two slices of bread as a single object, because the theorem only requires that the portion on each side of the plane vary continuously as the plane moves through 3-space.
No idea how this is related to the I Cut You Choose conundrum.