My Greek friend makes a similar pie with feta, spinach, eggs, onion, nutmeg and filo. Where the pie originates from - Italy or Greece - comes up everytime we share a piece of pie. I use puff or in fact any savoury pastry available at the supermarket.
Parmesan cheese is not truly vegetarian, as it contains animal rennet. To make this dish 100% vegetarian, omit the cheese or find a suitable vegetarian substitute made without animal rennet. In supermarkets look for the 'parmesan style hard cheeses' which are suitable for vegetarians.