The quick answer: it is not so much that part of your death star gets sliced away as that the death star as a whole will have an extremely hard time escaping even with maximum thrust, and the part below the horizon definitely will not escape. Either the ship tears itself apart or falls in.
Long answer: you approach the black hole along what normally would be a hyperbolic flyby trajectory. Sitting in the control room you do not experience any force, since you are in free-fall along a geodesic curve.
But as you get closer the structure starts to creak: different parts of the death star are at different distance from the black hole and subjected to different spacetime curvature. This produces tidal forces that try to pull and squeeze the spherical death star into an ellipsoid pointed at the black hole. You are however smug, since you choose a supermassive black hole and hence the tidal forces are not that severe, even at the horizon. Had you used a non-spherical spacecraft there would also had been a torque trying to rotate you, but again, death stars for the win.
Now you get very close to the event horizon, skimming it. The view outside looks decidedly weird (since you are inside the photon sphere and having relativistic aberration). At this point trouble appears: your free-fall trajectory does not continue back out to infinity, but spirals into the black hole! Unlike in Newtonian gravity close free-fall trajectories do not return out. The reason is that you are inside the innermost stable circular orbit (ISCO). The only trajectories in this region that do not end up in the black hole are sufficiently steeply outgoing ones.
So you turn on your super-thrusters and try to break free by increasing your velocity in the right direction. But now you have a problem: the part of the death star furthest from the black hole needs merely a lot of extra velocity to be on a trajectory that leaves the hole region. The innermost part needs to move faster than light to break free. So if you can achieve some maximum boost there are going to be parts that, had they not been connected to the superstructure but freely flying, would not escape. So your boost turns into stress across the death star. Something has to give, and no matter what boost you are giving the innermost parts through the engines and pulling metal they will not move outward since they are below the horizon.
It is at this point I strongly recommend running to the escape pods.
Exactly what the fate of the death star will be depends on a lot of things, but the simplest case is that the overall boost is insufficient and it all falls in. It might also break apart. Relativity shows that there are no perfectly rigid materials (they would need to have a speed of sound faster than light), so not even an imaginary death star will avoid getting pulled apart.
In the end, I leave you to explain your little joyride to the Emperor.
[It is worth noting that nothing looks strange locally when you cross an event horizon. It is a global property of spacetime, not something you can directly see. Extended objects crossing it merely have some parts whose future trajectories will meet the singularity, and the parts just outside may meet the singularity, but can escape if they break away somehow.]