How to repair walls which have two layers of wallboard?
I don't believe you have two layers there. I think that's what I had in my previous home, which is what I called hybrid drywall--a layer of 3/8" gypsum panels covered with about the same thickness of conventional plaster. The system was used in the 1950s as a transition between conventional lath-and-plaster and modern drywall. 

In my case the gypsum panels were 18" high and had heavily rounded, fully wrapped edges. I really liked it since it was extremely tough from a daily living standpoint but still easy to cut with a keyhole saw, say for new electrical boxes. 

I'd fur out your framing and install 1/2" drywall (or 5/8" for better stiffness, or cement board) flush with the original surface. I often used 1/8" hardboard or other furring and 5/8" drywall with good results, since the original wall was about 3/4" thick. It did vary somewhat, as plaster does, so check your thickness carefully. It's better to start slightly behind the original wall than to protrude beyond it and create a bump. 

It's a good opportunity to flatten any wonky framing, so use a long straightedge horizontally on your furring to check plane before you install the drywall.