Just tried Fina, on Fort Hamilton Parkway between East 2nd Street and East 3rd Street that just opened this past weekend. Although they do slices before 5 pm, it's not a brightly-lit greasy-slice kind of place. The decor shows that they're going for more of a Lucali kind of vibe rather than Famous Original Famous Ray's. We went after 5, when they only do whole pies (and, I guess from the menu, calzones).
Although they have some intriguing-looking toppings, we were interested in just trying the basic plain pie for our first dish there, and we liked it very much. The sauce is nice and piquant, the pies are doused with olive oil, and the cheese is good. I live close enough that I'll be coming by anyway, but I can see this being a destination place.