The Quiet Storm Coffeehouse & Restaurant seems to cover all of the bases. Breakfast, lunch, dinner, dessert, coffee, catering, and live music seem to make their list of offerings at the all-vegetarian diner. All vegan / dairy-free menu items (of which there are many) are easy to spot. They even have a selection of dairy-free shakes!
5430 Penn Avenue (at Graham)
Pittsburgh, PA 15206