The reversed-biased method is actually the basis of single-photon detectors used in quantum optics experiments, especially experiments with entangled photons. Here's a paper that uses exactly this--reversed-biased red LEDs-- to introduce physics undergrads to the essential concepts:
I read the paper; much of the electronics details got past me on the first read, but I think that the part most relevant to your question is their statement that it needs to be determined if the LED is a deterministic or a probabilistic single-photon light source.
https://advlabs.aapt.org/items/detail.cfm?ID=13808
It includes a simple circuit that probably anyone here on HN could build. (Even me!)