How does a brain split operation work exactly? Do they put a barrier between the two halves? Otherwise, I would expect that signals could still propagate: cut two nerves, then push them together, and I can imagine that a signal can still pass through; if not always, then in a percentage of cases.
What you say is theoretically possible if the neurons weren't living cells and humans had the technology to reconstruct the brain's wirings. Your body isn't going to figure out which "wire" connects to which on its own.