I put a handblown glass vessel sink in my new bath and after about 9 mo or so it cracked. No one would accept responsibility for it and I wasn't going to pay for another, just to have it crack again. (artist said it was installed too tightly, plumber said it couldn't have been any looser or it would have leaked...yada yada.) Since I'd designed everything in that bath to go with the sink, including the wall mounted faucets, and the original sink was quite shallow and not too large, I had a HECK of a time finding a replacement. But I finally found this one and I love it. I love it more than the original sink because it doesn't show water spots like the glass did. I like that it's only 3.5" deep and can also be installed semi recessed if you need it lower. It's 16.9" in diameter.

Here is the original, so you can see placement. I don't have a photo of the new one in place but will post a retailer pic. Don't know if it might work for you.

My counter is deeper than this one, but I don't remember the depth - not too deep. I can't measure right now but if this interests you, I can measure and take a photo of it in place.

Here is a link that might be useful: Round vessel sink