Unfortunately, the french doors don't have enough of a "lip" to mount anything inside the actual door. Hence my idea to put the rod just above, even grazing the top of the door frame/molding. The rod for the panels is mounted wide, and the panels are pulled open 90% of the time, so you'd never see the white rod where it extends past the doors. I wish I could photoshop something but it'd probably be faster to mount the rod in real life than for me to photoshop something!!

Yes, a finish carpenter would do just fine. In fact, anyone handy with wood working tools would be fine. It really is an easy task to accomplish. I would not remove any drywall, especially since you have 5/8". 5/8" is normally used for fire code purpose, commonly referred to type "X". Your local jurisdiction building department will gladly answer any questions to if you are permitted to mount any surface to replace drywall. Here, paneling etc must go over drywall. You can do a search for type "X" drywall, to find out where it's mostly installed. Just to clarify with what I posted earlier about running an extension on the back side of the casing. Should the BS be run as posted (stopped short of the rough opening of the window), the casing would set on top of the BS until the BS stops. Then the extension on the back side would be the filler of the gap of between the casing and drywall. The casing would be brad tacked to the jamb extensions, and could be dabbed with adhesive on top of the BS. I believe this would present the cleanest look. Of course I may have confused you by trying to explain! Ugh, always easier to do, than explain.

You might consider using a pair of shorter curtain panels that also ends inside the frame rather than falling out/beyond the window sill. A pair of curtain panels with a pocket for the rod rather than hole rings for the rod could also help. If your budget or your taste require you use the curtain you have now, consider washing it to make it less stiff and more easily folded or gathered between holes for the rod. If that doesn't work , you could add a valance covering the top and sides of the window frame. https://st.hzcdn.com/simgs/1461f6fc0762c757_4-1650/traditional-curtains.jpg or covering just the top https://s-media-cache-ak0.pinimg.com/736x/6b/30/b2/6b30b2bc087715b666526e1d81ba51bb--linens-and-lace-valances.jpg If you want long curtains or drapes -- longer than inside the window, you;ll probably need to put your rod on the wall above your window.

Thank you for all these suggestions! I measured the windows to see if roman shades would fit, but they're not quite deep enough, and I think outside mounted would block too much of the view. I guess I'm back to trying to mount drapes! The cafe curtains are lovely but some of the windows face neighbors, so I would need more privacy.