Fans of the Ham sandwich theorem know that any set of points can be divided by a plane into two equal halves.
Consider instead a 3-D shape that must be divided into 2 equal pieces by a single cut. A sufficiently bent spring washer or keyring cannot be divided into 2 pieces by a plane. But it's possible to make a simpler cut that works -- a partial plane cut.
Is that the simplest shape that cannot be split into 2 pieces by a plane?
Is there a simple shape that cannot be split into 2 equal pieces by a simple cut?
The Ham Sandwich Theorem says that given three measurable subsets of $\mathbb{R}^3$ can be cut into two equal (with respect to measure) pieces by a single plane. In particular, we can choose two of our sets to be empty. So any measurable subset of $\mathbb{R}^3$ can be cut in half by a plane.
EDIT: I originally misunderstood. You're interested in when the cut pieces are connected.