https://www.barkercabinets.com/Finished-End-Panel-APPLIED-DOOR-base-cabinets-p/bfeapplieddoor.htm

So, I’m assuming you’re going for an island similar to this older example:
https://www.barkercabinets.com/Island-with-glass-door-end-caps-s/260.htm

With a 1″ proud side cabinet, you’ll need to finish the back of that cabinet as well as the sides using full-height finished end panels. A paneled end panel in the 6.75″ to 7.5″ depth range is pretty shallow, but it is possible.

You can order using this product, but change the first option to “Squared Outside Edges.” This will give you wider top and bottom rails, which will look better at those smaller widths. This product is available in the 6″ to 8″ range.
https://www.barkercabinets.com/Finished-End-Panel-APPLIED-DOOR-base-cabinets-p/bfeapplieddoor.htm

You’ll also need to use a slab finished end panel to cover the exposed backside and prevent the plywood edges from being visible after installation.

If you’re using base cabinets for those 6″ deep sections, they will have doors on the face. The finished end panels should be ordered 5/8″ proud so they slightly overlap the cabinet doors, creating a clean 1/8″ step back. This detail is important, as it helps hide minor hinge adjustment variations and keeps the overall look clean.

I’ve put together a mock-up below showing the top view and how I would personally configure something like this.

another close up: