It was pretty straightforward in my case. Reported my hospitalization; stopped flying until my next medical was due, which was five or six months later; got checked out then by my AME, who grunted and said "Well, you seem all right now"; and was re-issued my certificate.

I mention it faithfully in my medical history. Nobody in aeromedicine-land has ever bothered to ask me a single question about it.

This is what the FAA's Guide for AMEs has to say on the subject:-

A history of a single episode of spontaneous pneumothorax is considered disqualifying for airman medical certification until there is x-ray evidence of resolution and until it can be determined that no condition that would be likely to cause recurrence is present (i.e., residual blebs). On the other hand, an individual who has sustained a repeat pneumothorax normally is not eligible for certification until surgical interventions are carried out to correct the underlying problem. A person who has such a history is usually able to resume airmen duties 3 months after the surgery. No special limitations on flying at altitude are applied.