'Breakthrough experience' is a loose term. The question is, what is the individual breaking through? In LP terms, breakthrough experiences seem to be a variable composite of Clearing Experiences, Awakening Experiences, and even Completion Experiences
White uses the term Breakthrough Experience in the context of his discussion of Transformational Change.
The term has some descriptive power in my eyes. If we imagine that the belief system - the Archetypes - constitute a puzzle of beliefs. Due to archetypal interpenetration, an attempt to change a specific belief will result in the resistance from all other beliefs. Due to archetypal interpenetration, the other beliefs/archetypes, "resist" the change. Of course, it is the PU itself that resists, since it cannot make sense of that specific change which does not fit well into the archetypal "puzzle". "Breakthrough Experiences" are experiences that are strong enough to push through the barrier created by archetypal interpenetration. Strong enough to lend sufficient credibility to the specific change in belief so that it can "confront" the story that is told by all other archetypes. Thus, a new "picture" emerges, as the mind accepts the new belief (hopefully closer to the truth) and quickly reorganizes COEXes to suite the new constellation of beliefs. The reorganization of COEXes may take time, and is certainly not achieved entirely during the Breakthrough Experience. However, due to the higher CQ, neuroplasticity ought to be very high, making such experiences possible in the first place.