There is this really great place on 41 rue des Bains called Tavolone. It is Italian of course and it feels as such. They have great wines, great food (pasta, salads, paninis, piadines, etc). Check out the menu. You will need to reserve, although there is always a way to find a place.
It is very vegetarian friendly (I am vegetarian).
Tavolone: 022 321 12 12 / 41 rue des Bains