Detecting electrons as they tunnel through the band gap of a P-N junction
It's easily done. Reverse-biasing the emitter-base junction of an NPN transistor is well-known as a cheap and fairly good noise generator. See http://holdenc.altervista.org/avalanche/ as an example, but if you Google reverse bias noise generator you'll get lots of hits.

And the problem is not so much amplifying the random electrons as it is reducing the number to a level where counting single electrons is possible. Actually, this isn't done. Instead, the standard approach is to AC-couple the noise voltage and look for zero crossings of the detected signal.