You'll enjoy that this reminds me of gelson's, the high end grocer in Marina del Rey that was the closest grocery to my house and job. I mostly loved it though I had to choose carefully lest my wallet be stripped, good food all around. (The other choice was lucky's, anext the barrio, where even the potato chips were burnt in the bags.) Back on gelson's, they did have an actual pizza in their deli with sliced carrots on it (rolls eyes). You're talkin with a purist here..
On the other hand, they had fabulous spinach cheese things in dough.. would that I had the recipe. I now forget the name, but I tried to contact them for the recipe and the deli folk had changed hands, or maybe menus..
plus it was the only place I've ever seen that made fish sausages (sunfish and spices, if I remember).