Although the Pink Flamingo isn't a vegan restaurant, I eat their pizzas often enough to owe them a post.
There are four franchises in Paris, one of which happens to be on my street, and another (the original) on a good friend's street. The pizza bases are 100% vegan, and made with organic flour. Your best bet is the Aphrodite pizza sans fromage. This results in a perfect pizza base topped with tomato sauce, fine slices of grilled aubergine, garlic, red pepper flakes, and a big dollop of hummus in the centre to dip those pesky crusts in.
The restaurants themselves are quirky, maybe overly so. Think pink and black colour scheme, Ramones on the stereo, a pizza called the Obama. If you choose the delivery option, you'll be given a pink balloon so that the delivery cyclist can easily spot you and bring you your pizza. See, quirky.
Your non-vegan friends will have a lot of unusual pizza toppings to choose from, the Indian-inspired Gandhi being popular with vegetarians.
You save a euro per pizza if you choose take-away rather than delivery.
Addresses in Paris:
23 Rue d'Aligre
(closed Mondays, like pretty much everything on my street)
67 Rue Bichat
(the original branch, near the Canal St Martin)
105 Rue Vieille du Temple
(the branch in the Marais)
30 Rue Muller
(the branch in Montmartre)