Can we split an electron (or any other elementary particle) like we split a photon? On this site, there are many questions and answers about splitting a photon. Most of them right away clearly state, that the term splitting is not quite right, because in most cases, the original photon is absorbed, and then new photons are emitted, and the sum of the energies is what is the same basically. 
Can you split a photon?
Can a photon be split?
The methods are different for splitting a photon:


*

*SPDC

*pair creation

*strong magnetic fields
Now photons are unique elementary particles, because they are the only free particles with no rest mass, no EM charge (gluons are in confinement, gravitons are hypothetical).
The only thing I found was about electron fractionalization.

Specifically, what SSH showed is that when an electron
  is added to an otherwise neutral polyacetylene chain, it
  can break up into two pieces, one of which carries the
  electron’s charge and the other its spin

https://arxiv.org/abs/cond-mat/0106126
I have not found of any other elementary particles that could be split in any way, my question is whether we could split any other elementary particles like we do with photons.
Question:


*

*Can we split an electron (or any other elementary particle) like we do with the photon?

 A: A glib answer to this question is to reply dismissively about lepton number conservation --- a symmetry which electrons obey but which has no equivalent for photons. You can't turn one free electron into two free electrons, because the total number of leptons (minus anti-leptons) in an isolated system is a constant. (You don't seem to be asking about electron-positron pair production, nor about weak interactions where neutrinos get involved in the lepton sector.)
However, your link reminds that, in matter, there are many indistinguishable electrons gadding about. With many excitations of the electron field present, a careful experimenter can construct a system where the number of interesting electrons is changed in some interaction, while the number of background electrons in the ocean of whatever crystal lattice you're using is not significantly changed.
For a pop-science level discussion of condensed matter systems where you can talk about collective excitations of matter the way that high-energy physicists talk about particles as excitations of the vacuum, you might like to read Laughlin's "A Different Universe."  It's not directly related to your question about splitting electrons, but I think Laughlin's perspective might help you understand better what it would mean for "an" electron to be "split" in an interaction with a big molecule. 
This was more of a comment than an answer, but it got too long for the comment box. Perhaps it will inspire a condensed-matter person to give your question the thought it deserves.
