It would appear theoretically possible (to some degree) through extreme applications of recycling to trigger mass shedding in pulsars.
Pulsars are rapidly spinning neutron stars, the fastest class of which are millisecond pulsars. The current belief is that they build up rotational speed through accretion, a process known as recycling. One study, Recycling Pulsars to Millisecond Periods in General Relativity (Cook, et al), explores the limitations of this process.
The following chart shows their results:
At the point where the dotted lines meet the two plots, you can see a reduction in mass at those energy levels. This is due to the angular velocity of the body creating instability which results in mass shedding - essentially mass at the equator of our neutron star being flung off the star due to the body's angular velocity.
Unfortunately, this is not exactly an easy process.
The timescale to accrete the required rest mass, ~0.1 M☉, at the Eddington limit, ~10-8 M☉yr-1, is ~107 yr. This timescale is largely insensitive to the adopted nuclear equation of state. If other astrophysical considerations require a considerably shorter time scale, then the simple recycling scenario described here will have to be modified beyond the variations explored in this paper.
(Note however that the research here is actually attempting to avoid such instabilities, and they accomplish this by adding even more mass, such that the body can support even greater rotational velocity without encounter instability. Additionally, they're trying to create millisecond pulsars, but we don't need to do this as they exist naturally, so we could save ourselves a lot of time by (very carefully) approaching an existing millisecond pulsar)
I don't think this would exactly be breaking apart (despite Wikipedia's use of that exact verbiage to describe it), but it allows for the return of mass that was at one point in a neutron star. Of course, chances are our theoretical neutron star miners are very likely to be the ones who put that mass on the neutron star to begin with. On the other hand, this (hopefully) accomplishes the task without reducing the object to a quark star or black hole.
Cook, G. B.; Shapiro, S. L.; Teukolsky, S. A. (1994). "Recycling Pulsars to Millisecond Periods in General Relativity". Astrophysical Journal Letters 423: 117–120.