Is there any experiment where QM can predict individual results? This is my question:
Is there any actual experiment or observation that only needs to be performed once (rather than repeating it a hundred times and analyzing the ensemble), the result of which can be predicted exlusively by quantum mechanics?

This is the reason for my question:
From what I've learned about Quantum Mechanics, it is a model that can predict the results of multiple identical experiments. The more repetitions, the closer they will come to QM's prediction. However, if you do only one experiment, then QM can't predict the result.
So that got me thinking that maybe QM can't be considered a useful model for single experiments, it's only meaningful for ensembles of experiments. When one tries to give an interpretation of what's 'happening internally' according to a model, it is only correct to do so if the model itself is meaningful. And just how meaningful is QM when it comes to individual observations?
For example, in the double slit setup, is it really valid to try to interpret what happened to a single electron using QM? "It went through both slits", "Its position is not defined until measured", "It interferes with itself". All those statements come from trying to interpret what happened according to QM, but I don't think doing that is really valid. QM makes (almost) no predictions on a single electron so one can't try to interpret what happens according to it. It is no more valid that trying to interpret it using Newtonian mechanics or even Aristotelean mechanics.
I don't want to ask a vague question, so I focus on asking to confirm if there's any situation that corroborates QM without needing to have an ensemble of experiments. Like, the heat capacity of some metal, or the spectral lines of some element, etc. An experiment that always gives exactly the same result individually.
 A: The GHZ experiment is one that can give a result that is incompatible with locally-real theories with a single measurement. In this sense it is an improvement on the Bell experiments, which have to be run multiple times to and present statistics to demonstrate non-classicalness.
Basically, if you get three particles entangled in the right way and then perform just the right measurement you the classical and quantum predictions contradict each other no matter what, not just statistically.
A: I believe this question is a subset of a more broad question: "When we speak about very small pieces of matter can there be performed a single experiment with only a single isolated particle to test some theory about it". And my humble guess is that the answer is: almost definitely no.
The little bastards are always on the move and the smaller they are the harder they are to trace with reasonably limited amount of interference to whatever they intend to do and there is always a lot of other matter around it to interact with. So at some scale any experiment inevitably comes to a point where every single particle pretty much does whatever it happens to like at the moment and there is absolutely no way to know for sure what will happen to it. And for many practical purposes you should expect to end up in the state well before you start to feel the need to dive into quantum mechanics or some other weird stuff. You will need to use statistics in your experiments just because the temperature is not 0K, air pressure is not 0, the stuff you are testing is never perfect and you overall just have too much various random junk flying around. In fact most of your experiments probably need some data processing to provide at least some kind of an estimate of how reliable the results most likely are.
A: I will address this:

QM makes (almost) no predictions on a single electron so one can't try to interpret what happens according to it.

In dealing with probability theory, it is a truism that a true die will have a distribution of throws for the numbers from 1 to 6, equal in height within the statistical errors of the throws, the more throws, the smaller the error.
Suppose you take a die and throw it 2000 times, and instead of a uniform distribution there is a peak at number 6. Is that not a measurement? What does it tell you? That the die is biased towards 6.
In the double slit experiment a single photon at a time through the slits, looks random, starting from the left


Single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames.

On the far right the pattern predicted by classical optics is reconstructed.
This tells me that the distribution of the photon particles  is not uniform, but biased with the expectations of classical light.
Quantum mechanics is the biasing  factor.
With electrons, it is even worse, because there is no classical mechanics theory that can predict that the seemingly random accumulation of electrons  ends up with wave interference patterns like the photons.

So quantum mechanics is necessary to explain these experiments.
