We found Petrino from a recommendation by our host. My wife is gluten free and we found Petrino's menu to be extremely accommodating. All of their starters are gluten free as well as a majority of the remaining menu.
As for the food quality, it was nothing short of exceptional. We ate at Petrino's multiple times during our stay on Naxos. Every meal was done perfectly. The meals were full of flavor, plated beautifully, and described by the staff as they were delivered.
Lastly, the staff was phenomenal. Always willing to provide suggestions on what to order, answer any questions we had around gluten, and extremely friendly and inviting. They made us feel like family.