Curved Doors and Drawers

I am attempting to make the Sheraton sideboard shown in plate 66 of Verna Salomonsky's "Masterpieces of Furniture." I have never made curved doors before. I think I will make the top circular drawer by laminating thin pieces of mahogany. It is only about 4.5 inches wide so I can resaw this easily. Can anyone describe a general method for making the other cabinet doors and the cellerette drawer fronts? These are approximately 13 inches high. I would not be able to resaw that height. Would I glue up strips of a secondary wood like poplar with the grain in a vertical direction, offsetting each one to make a "stairstep" like curve and then smooth them out with scrapers and spokeshaves? Or should the grain be in the horizontal direction and built up with segments laid like bricks overlaping each piece and then smoothing out? I think I would like to avoid plywood or MDF like materials. Thanks
It was normal practice to brick-build curved drawer fronts, but curved doors – as on sideboards, bow-front corner cabinets etc. – were made up from vertical, bevelled strips of wood, say, 1-1/2" to 2" wide and then faired to the curvature required.