What you are wanting(custom made motherboard) would be extremly expensive. Design, testing, retool, production and testing. You'd likely be looking at 10' to 100's of thousands of dollars and thats if you could even find a motherboard manufacture willing to do it.
I think you're missing the point. "Just for one" makes it even worse.
Manufacturing is based on volumes. Not only would you have to get a schematic for the board, you'd have to find someone with the equipment willing to divert it to make one for you, acquire the materials....
As above, probably several thousand dollars, at minimum. Your best option is trying to find a refurb board or units being sold as "non-working, for parts" on eBay. If you're talking about making a board that would take a new socket but fit in your current chassis..... that's not happening. Simple as that.
If you are willing to drop money on a custom motherboard for your laptop, then you might as well purchase a high end gaming laptop from a boutique laptop brand like Origin PC. It will be far less expensive to do so.