(Caveat: performing Instant Autopsies does not take away the ability to select the Breakthrough. (Yes, this includes that you can be offered a Breakthrough immediately after completing a Breakthrough, though it's uncommon) If you do not immediately start on the Breakthrough, it will promptly vanish.

In short, each time you complete a Research project ( Not a Shadow Chamber project), there is a chance that a Breakthrough will be offered. Breakthroughs are yet another ambitious mechanic added in by War of the Chosen, this time connecting to Research.