You could use a router and a template or pattern bit. This video from the Wood Whisperer shows a similar technique for making bowls using a router, although you would want a straight (pattern) bit rather than the round-nose bit. He hogs out most of the waste in the center with a Forstner bit on the drill press, although you could do the whole thing with a router.
For this method, you would need to buy or build a circular template of the correct size (the inside diameter of the template would be the same as the diameter of the black area in your sketch). Circular templates are readily available for purchase in smaller sizes, but you may need to make one for your ~12" piece. A sheet of 1/4" plywood / MDF would be perfect, and you can use the router to cut out the circle here as well. Circle-cutting jigs are available for purchase, or you can make one yourself (lots of videos online, here is an example from April Wilkerson)