Uroboros -
there is a thread here that explains installing UCL and wiring to a switch. You can put the transformer in basement or attic or in a cabinet where it can be accessed. We put ours in the cabinet over the microwave but then ours are not wired to a wall switch.

also - ours are 1/2" thick.

the link to UCL install...

Here is a link that might be useful: UCL - DIY

You can also mount a power supply (AKA 'electronic transformer') on the bottom of the cabinet.
Many are less than 1 inch thick.

The problem with remote transformers is that any wiring concealed in walls must be an 'approved method.'

Most of the time the smallest gauge you can find is #14, but if you have a high capacity transformer you are going to need larger wires.

Mmmmm

So I would have to plug the transformers into an outlet.

And have the outlets connected to the wall switch.

Is that correct?

Would I need two separate transformers for the two cabinet sections?

Some equipment is available and listed for hard wiring without a receptacle.

If you use lights that require a receptacle it must be inside the cabinet and not in the counter back-splash.

Permanent lighting is also not allowed on the kitchen counter small appliance branch circuits.

It can take up almost no cabinet space if an electric box is recessed through the back of the cabinets into the stud cavity of the wall.
The receptacle plate is flush with the inside of the back of the cabinet.