Learn Italian: Avendo analizzato la malattia, il medico ha preparato il vaccino. - Having analyzed the disease, the doctor prepared the vaccine.

The past gerund (avendo analizzato) is used to show that the action happened before the main action (ha preparato). It also strongly implies a cause-and-effect relationship: because the doctor analyzed the disease, he was able to prepare the vaccine. If we used the present gerund (analizzando), it would mean the actions were happening at the exact same time.

Could I use avendo analizzato if a scientist analyzed the disease, but a different doctor prepared the vaccine?

No. In Italian, a gerund clause usually requires the same subject as the main clause. Since il medico (the doctor) is the one who prepared the vaccine, it is understood that he is also the one who analyzed the disease. If the subjects were different, you would need to spell it out using a different structure, such as Poiché lo scienziato ha analizzato la malattia... (Since the scientist analyzed the disease...).

How do I know whether to form the past gerund with avendo or essendo?

You use the exact same auxiliary verb that the verb takes in the passato prossimo (present perfect). Since analizzare takes avere (ha analizzato), its past gerund is formed with avendo. If the verb was a verb of motion or state that takes essere, like andare, you would use essendo (e.g., essendo andato).

Is there a more common, everyday way to say this without using a past gerund?