Got pictures?

What do you mean primed wood counters are best left without anything except oil finish I am more concerned with the actual working with this the doors are really not thick enough without some sort of base and where they are joined is going to be a trap for all kinds of gunk. FYI wood has an enzyme that actually kills bacteria but once you seal it that enzyme cn’t work . I have butcher block in my houses for years they always look new just by re-oiling when needed and a light sanding if a stain or burn occurs

I am concerned about water damage, since they are in the kitchen.

If the doors have a primer on them (usually white), you will need to strip the primer off to get to the natural wood. Then you can stain them and finish with Waterlox or something similar. Primer is usually applied before something is painted.