We had to take down several large sections, about 6x4ft sections, in order to run updated electric. The problem is that the 3 layer plaster is 7/8in thick. What is my best solution to put up new even thickness drywall? Most of it will be hidden under our new kitchen cabinets.
Shim out the framing to get the drywall flush. I've often used 5/8" drywall with no shims, depending on the plaster thickness and patch configuration, but otherwise shim it.
1/8" hardboard is cheap and easy to work with. Or rip some scrap lumber to size. Staple it in place and be happy.
Update: After seeing your photo, I suggest re-cutting your edges with a circular saw and abrasive blade set to depth. You really want a nicer shape to work with. Snap or trace straight lines, level and down the center of studs. Use ear, eye, and respiratory protection.
Agree with the advice to remove, insulate, and replace.
Couple of notes: add nail plates generously where wiring/plumbing is less than 1.5" from the surface of the stud.
Especially if you're tiling (and less so if it's just paint), pay attention to how straight the wall is at the bottom, middle, and top. Shim or shave as needed to get your studs all in the same plane.