Fix pickled doctrees cache.

[picnixz]

Closes #11474. Closes #11908.

Note that this fix may increased build time for very large projects since we need to revert back to reading at least twice the pickled files.

Alternatively, we could add the possibility to mark which document should be 'refreshed' from its file if needed? though I don't know which ones we should actually target. Maybe @AA-Turner has some solution for this?