Double-sided Grill?

My esteemed friend Blork just made grilled pizza.

As far as I know, it's quite difficult to pull off a pizza in a home grill. Add to this a circumstance of having to have a tiny grill for your balcony because they're forbidden in your apartment building, and you can imagine the conundrums associated with my even considering grilling a pizza.

The damned thing is, and you can look it up, the bottom of the pizza burns long before the top is done. You can imagine the result -- charred bottom, half-melted cheese.

Now look, don't hassle me here. Yes, there are workarounds. Oh, yeah, you can cut holes in your grill lid blah blah blah but I'm not a great go-to DIY guy. I don't like blowtorches, either.

I was thinking more along the lines of a miniature wood-fired oven, except in your grill. If you read Blork's subsequent comments, you'll see a solution that involves convection, but I'm more convinced that a direct-heat method would be better. But how? I'm thinking either of some form of stone built into the grill lid (hmm, heavy) or some kind of grille built in that you would layer coals on on aluminum foil. That way the ashes wouldn't fall on your food but the heat would blast the top of whatever you were cooking.

Convection would work, but it wouldn't GRILL, rather steam . . . I'm not a kitchen tech dude, so I don't know if this would be the case, but it seems that would be the case.

So: problem: come up with a small, balcony-sized grill that has a powerful heat source in the lid as well as below the food, preferably charcoal. Make it easy to assemble and not a danger to fire up.

Anyone listening, Mr. Inventor?