Rather than tearing up a cabinet box, since it sounds like this is aesthetic, just pad out the cabinet.

IOW, rather than screwing something directly to the wall, put a board, 2x4 or whatever your desired projection is, and hang the cabinet on that. Keep in mind the cabinet edge on the side usually has a tiny lip. You'll have to put something in there to keep the cabinets spaced to each other, since usually you screw these face frames together.

As far as making them taller? Again, rather than chopping up a good cabinet and compromising its looks and integrity, stack something on top.

For example, I needed a 21" tall wall cabinet. Rather than spend $$ on a new, special cabinet, I took a regular wall cabinet and cut it down to 21". Because I'd be looking up into it, I put the finished bottom as the ceiling of the cabinet.

On another, I did the same thing, but because I wouldn't be looking up, but could rather see the bottom of the cabinet, I put the nice end down and the open end up. I had another cabinet on top of that, so it worked out fine.

I forgot to mention that if you buy a 2x3/4" piece of wood, it's actually 1.5", so it's the size of a normal face frame piece. Just screw one in there. I use my handy-dandy Kreg pocket hole jig.

So yes, you can do it. You'll have to be creative.
Just ideas.

Good tips. I'll run these by the handy man and see what he can do with this.

Thanks!