Shiplap, beadboard, dummy doors as suggested. Paint it as an accent.

There's different things you can do. Matching the same as the front is one. You could put crown moulding underneath the couter and trim along the bottom. Or decrotive paneling. You definitely need to do something about the base and corner shown in the picture. You can see the nail holes! Beautiful kitchen otherwise.

we have to find a finisher that will show up! The front of the island has dark wood drawers and the opposite wall has a darkwood cabinet with the same backsplash as shown. We have stainless toe kick molding that will be all around ...soma is on already and looks nice. Would white cabinet doors look more modern than a dark cabinet door for back of island?

this is another view in daylight

Both colours would definately work. I believe white would give a more modern look while dark wood would give that classic blend -while still having the modern feel -(example below of similar to your situation). I havent seen a picture of the front or sides of your island, but what ever colour you deicide I would finish the sides as well. That being said white could look really good so that its all uniform until you walk around to the front where the dark doors would really pop.

How about putting your backsplash there?