That's an unexpectedly neat question.

Gravitational radiation can form caustics, so in one way, yes. This was an area of theory that Bondi and Pirani were interested in during the 60s to 80s. Caustics are meant in the sense of the bright spots in
https://en.wikipedia.org/wiki/Caustic_(optics) but rather than some "rogue wave" you might get a black hole.

More recently mathematical-relativity studies of the stability of Kerr black holes included a 900+ page paper https://arxiv.org/abs/2205.14808 which among other things considers something like a rogue wave passing through a spinning black hole, and asking: does the black hole swallow up some of the wave? or does part of the wave get entrained into a sort of accretion disc? in that disc do caustics appear, and if so do they lead to black holes? if so, do those black holes fall in, fly away, or coalesce into long-term orbiting bodies around the original black hole? These questions were asked because "stability" means "if perturbed (e.g. by a powerful gravitational wave), does an initially Kerr-like black hole relax back into a Kerr-like black hole, or does it become something very much not like Kerr?".

Caustics may also form by a powerful gravitational wave gravitationally-lensing around some massive object like a heavy galaxy cluster. And if that massive object has enough angular momentum, it's possible the caustic will evolve into a significant gravitational wave enhanced somewhat analogously to a slingshot manoeuvre by a spacecraft.

Conversely, we can start with a rogue wave existing, and see what it does.

Bondi and Pirani were also interested in spacetimes with enormous gravitational wave impulses. One family of those is the "sandwich wave", https://royalsocietypublishing.org/doi/10.1098/rspa.1959.012... (1959) (put "https://sci-hub.se/ in front of that URL if you need to).

Loosely, it's called "sandwich" because the spacetime is pretty bland (like white bread?) or even flat on either side of the wave, which divides the spacetime into three parts (including the wave itself).

Thirty years later they showed that the sandwiches are pretty unhealthy as they tend to destroy everything in the universe they pass through (any set of finitely-but-widely-separated galaxy clusters in such a spacetime will end up colliding with each other in finite time).
https://royalsocietypublishing.org/doi/10.1098/rspa.1989.001... (also on sci-hub, and with nice late-1980s diagrams).

Fortunately, I don't think anyone has any idea how to generate a sandwich wave - it's just a feature of the spacetime put in by hand.

Unfortunately, we could be in a sandwich spacetime and not know it yet. A sci-fi writer might speculate about a preferred orbital plane for a large number of (very-early-universe-born) supermassive black hole binaries building up an approximate sandwich wave over the course of billions of years. Or perhaps speculate that the apparent stochastic background that's the topic of the fine link at the top where I guess you get "churning" from is not caused principally by supermassive black hole binaries but by some tensor-mode reheating during cosmic inflation (or a false-vacuum-to-real-vacuum first-order (not as in recent Star Wars, or is it?) phase change), and that this effect happened to generate a sandwich wave which has already destroyed galaxies far far away. A hard-working hard science fiction writer might even be able to force a cosmologist to concede that such a picture might be consistent with observation, including the pulsar timing array results in today's news.

A sandwich wave would zip through our galaxy at the speed of light, but the sandwich-wave-induced collapse of the galaxy into a caustic might take enough time that you could really what's-the-opposite-of-enjoy the experience. Pleasant dreams.