I wasn't thinking of birthing method, but I guess that could work, too, as long as you make sure you end up with a very straight edge on the fold.
To avoid that challenge, I would finish the edges of the curves first (see linked article -- it's a gathering stitch; I'd make a circle out of a cereal box the size I want it to finish, then add 1/4-1/2" to be turned under. Here's a tutorial on the process -- just substitute your own cereal box template for KKB's mylar templates:
https://www.youtube.com/watch?v=sGEuyb7WNHM). Then, I'd pin baste the right side of the brown circle to the wrong side of the red square-circle. Fold on the straight edges & then top stitch around the whole thing -- especially the folded over brown fabric. Once you've done that twice, repeat with the fabrics in reverse order (red circle on bottom, brown square-circle on top). Then, just seam them together along the straight edges.