like this one

or this one

Sure you can. I am not sure if you are getting a large
sink like poorowner's first image. Maybe if you spoke
with a woodworker they could build a counter around that.
Not sure though.

Meanwhile check out these other sinks below. They all
have counters around them. Not wood but undermount sinks.
~boxerpups

This one is undermount look carefully

We have an 1840 farmhouse kitchen with wood countertops everywhere. We have black Waterloxed countertops on the perimeter with a 36" undermount farmhouse sink, and natural Watco'ed beech on the island with a 24" undermount prep sink, and have had absolutely no problems with damage from splashes, standing water, stains, you name it. Our kitchen gets regular hard use from multiple cooks, and it's doing great.

So if a wood countertop and traditional undermount sink fits with your period home and design aesthetic, why try to force something else? We have that set up in two different locations with two different wood finishes, and it doesn't cause us a minute's heartburn.

If you want to do the farmhouse sink undermounted with the backsplash, it's unorthodox, but you probably can (with a flexible and competent woodworker on board). The fussy part is going to be the countertop cuts at the back corners of the sink. They'll have to curve pretty exactly around the backsplash to come straight out along the sides, and be well caulked.

It can be done, and of course, if you want to spend the time and money to have it done, it's your choice. But if your only motivation for the unorthodox choice is the durability of the wood countertops, I'd say don't worry about it , get the traditional wood countertops and plain farmhouse sink, and spend your money on something else totally fantastic.

Have you found the sink you would like in your kitchen? Have you seen such a sink undermounted? It would be very difficult, but possible, to undermount the clarion sink.

I have the Clarion Farmhouse sink pictured above with the continuous backsplash. It is a nice sink for my uses, but I don't think I would want it for the only/primary kitchen sink. The drainboards are 18'' . They will accomodate only a small dish drainer.

My clean up sink is an undermounted farmhouse sink with a wall faucet. The sink sits enough forward on the counter that the only the spout of the faucet reaches over the sink. Because the Clarion sink's wall faucet is mounted directly above the sink, it limits the vertical space in the sink. In fact, I need to replace the spout with a very short spout. As it is now, the spout reaches beyond the sink!

If you would like to see pictures of my sinks, let me know.