Could you please tell us about the advantages of leaving the beard (One handful or more) and shaving the mustache according to medical science?

All perfect praise be to Allah, The Lord of the Worlds. I testify that none is worthy of worship except Allah and that Muhammad is His slave and messenger.

We highly appreciate your interest and keenness to be knowledgeable about what is lawful and what is unlawful, and your enthusiasm to learn about the principles of your religion.

Our answer is that whoever is aware of the rulings of leaving his beard to grow and the relevant Islamic etiquette concerning trimming or shaving the moustache, then he has to abide by those rulings and must not neglect them. As for the medical aspects (advantages) pertaining to growing the beard and trimming the moustache, you may consult a knowledgeable doctor.
As regards a Mufti (jurist/scholar), his duty is to state the ruling on that matter. But on the whole, it is to be taken into granted that there is nothing that Allah Almighty enjoins us to do except that there is a benefit and advantage for us in doing it. On the other hand, there is nothing that Allah forbids us from doing except that there is harm or disadvantage in doing it. In prohibiting and forbidding, there is surely a Divine Wisdom, which we may either discern, or not. A Muslim has to submit to the will of Allah in all cases, whether the wisdom behind a given ruling or command is known to him or not.

Allah Says (what means): {But no, by your Lord, they will not [truly] believe until they make you, [O Muhammad], judge concerning that over which they dispute among themselves and then find within themselves no discomfort from what you have judged and submit in [full, willing] submission.} [Quran 4:65]

For more benefit on keeping the beard, please refer to Fataawa 332320, 317029, 119302, and 91108.

Allah knows best.