Here is a photo of Moser's "Ellipse" stool.
The seat is solid stock, shaped with a saddle depression as shown.
Stretchers are tenoned into the legs, and I am wondering if the leg mortises are done as a shallow round bore, then a square mortise inside. This, to eliminate the work of doing shaped shoulders to match the curve and taper of the legs.
The larger top stretcher-supports have small span supports between, two of them, which support the seat. I think three screws each, the two outermost ones in slotted holes, up through these to fix the seat on.
I threw a preliminary doodle of this up on the 3D Warehouse, after doing a study of the 25" high version, using Sketchup. My model is missing the two cross members under the seat.
For machining the thick blank for the seat, hogging most of the material away before going after it with a grinder or hand tools, I envision a couple fixtures and a router sled. Fixtures also needed for machining the mortises into the legs.