I like that. If the drawer is not an absolute must have, a couple of 2X4s equally spaced and secured from the top to the bottom of the desk top to serve as joists ought to provide all the support necessary. Joist hangers attached to the sides of the desk would further stiffen up your cook center.
Thanks to all for the comments.
I would like to keep the drawer,
There is a stiffener for the rear edge to prevent it from sagging.
Actually, part of what we see is an illusion. I put a four ft straight edge on the top and the sag is about 1/8 inch.
Since I will not be setting at desk I'm not worried about leg room. I think I will just make a 2X4 leg for the front center.