Obviously, cutting a neutron star in half would be pretty difficult. But let’s say you could - maybe you hit it with another neutron star traveling at a good percentage of light speed.
Or, let’s just say that you were able to remove that proverbial teaspoon of pure neutrons that keeps getting compared to the weight of Mount Everest.
What happens then? The weight of the chunk of neutrons is too low for a neutron star to have formed in the first place. Does this mean that the material emits a huge burst of beta radiation and reverts back to standard atomic nuclei? Or, is it more like a black hole which can lose half its mass and remain a black hole? Having compressed the material so much, is it stable as a mass of neutrons?
It depends on how massive the neutron star is. The size of a neutron star is a balance between the squeezing force of gravity and the expanding force of the nuclear matter, which would rather be protons and electrons than neutrons. There is a limited mass range for stable neutron stars, from about 1.4 solar masses to 3.2 solar masses. A larger mass will collapse to a black hole. A smaller mass would blow up to be a white dwarf, made of protons and electrons and stabilized by degenerate electron pressure. So if you cut a 3.2 solar mass neutron star in half, stand back and wait for it to settle down to two 1.6 solar mass neutron stars (assuming with no justification that it didn’t eject any material). If you cut a 2 solar mass neutron star in half and separate the pieces, neither has enough mass to avoid blowing up to white dwarf size.