I've used this penetrating epoxy outdoors, successfully, although on relatively soft wood:

http://www.epoxyproducts.com/penetrating4u.html

I don't know how far it would penetrate into walnut. As long as the area is out of sight, you might think about topcoating a penetrating primer with a tough paint of some kind. The web site above sells some marine/industrial paints. The proprietor would probably be happy to recommend something.

Be sure to finish both sides or your counter will cup. I've used a bar top epoxy (linked) that has held up for about 7 years now but I don't think that's what you're looking for (too thick).

Here is a link that might be useful: AeroMarine bartop epoxy

I see value in sealing the end grain of the sink cutout particularly well. If you thin the epoxy with its appropriate (often proprietary) solvent, then use a low heat gun, it can get into the end grain quite deeply. But don't burn the epoxy.
Casey

Use 100% vinly tape for masking off the surface you want epoxy-free.

Epoxy is self levelling. It will run over the sides and drip off the bottom, leaving little drips. Plan on masking these and removing the tape when the epoxy is set up, but not cured...usually about 12 to 18 hours later.

You can work the epoxy into the side and end grain by brushing it when it is flowable. Multiple coats are necessary. The cool thing about epoxy is it can be sanded away and the next coat bonds seamlessly.