We've just spent a great weekend on Texel Island with my fiancee. At this time of the year (November - off season) the summer-holiday island was clearly deserted. There were only a handful of good restaurants open, but we've been lucky enough to find the Vogelhuis Oranjerie in de Koog.
Food is great, the restaurant looks really neat, and there is a central fireplace to add to that. The waiters are really friendly and adwise you based on your likes vs. trying to offer you the priciest dishes.
Try the starters, with some extra helping of a side, if you're really hungry. These straters are huge, nicely presented and offer a local touch - shrimps / lamb / cheese from the region.
The prices are also the best compared to the other restaurants around. A must try, if you're on Texel.