It isn't often you find a quiet place in the middle of Paris, much less one with excellent food and outdoor seating. So the Restaurant du Palais Royal is special. The food is good modern French with interesting spicing and the wine list has some good finds hidden away (like a 2000 Gevrey-Chambertin, yum). But the real appeal of this place is the outdoor terrace on the garden of the Palais Royal. The Palais is now private businesses, but in the middle is an amazing and quiet formal French garden that is essentially silent in the evening. Makes for lovely outdoor dining.
It's a bit spendy but the prices seem quite reasonable for the quality of food, service, and the view. You'll definitely need a reservation to sit outside. If you're going for dinner ask about when they close the garden; some nights it closes at 10pm, making for an awkward transition during dinner. But last night it was open until 11:30 for perfect contentment in the gardens.
Thanks to Fabrice for the recommendation