This is one of those things that is best created with plain old shop equipment.
The last time I did one of these IIRC, the drawer front is cut extra thick. The joinery is all done while it is still flat (that would be things like dovetails). Then the bandsaw roughs the curves in. Any rabbets can be cut using a router table and a rabbeting bit. If a slot is needed, some idea, just use a grooving bit with a large bearing.
Watching something like that slowly cut on a cnc router and then knowing you STILL have to sand the heck out of it doesn't make a lot of sense to me. Now, if you do not have a bandsaw...that is a horse of a different color.
Edit: Whenever I use the cnc I always remember the great old saying "Just because you CAN doesn't mean you SHOULD".
Low Profile CNC Router Vise