I thought I posted the result of my chat with support for completeness.
Basically they told me that it is possible to do what I was looking to do. They would need to take the menudelgiorno directory out of public_html and I would need to change the configuration of the blog (it's a table in the database, don't remember exactly which one at the moment). To do that I would need to raise a ticket.
So it is technically possible. In the end though I have decided to leave things as they are. That's becuase I will most likely creare new blogs and I don't want to raise a ticket every time. The fact that the blog can be access by menudelgiorno.troccoli.it and by
www.troccoli.it/menudelgiorno is a minor glich I can live with.