The house was built in 1949. We are doing a whole house renovation. In the process, we cut out a piece of the wall when trying to move an outlet and discovered a solid 3' wall that appears to be a layer of sheetrock or plaster with a plaster like material about 1.25" thick on the inside. The house has two walls like this, 1) between the living room and smallest bedroom and it runs perpendicular to the outside wall of windows. The spacing between the windows(they are large modern floor to ceiling windows) is only 3-4" so we believe that is why they used a nontraditional wall with studs. 2) It is used as the back wall of two back-to-back closets. That's where the problem comes in. We need to hang heavy closets components on the wall in each bedroom. We're trying to figure out how to hang the closet system on the back wall in a way that will support shelving, rods and clothing.

What you could do is add another layer--but this time cover the walls with 3/4" plywood. Do the whole back wall. Sit the plywood firmly on the floor (not carpet). Screw it to the walls with appropriate anchors. Then, on the tops and sides block it with trim attached to the other walls and ceiling. You need to prevent it from pulling forward; the plywood itself is strong enough to support the weight of the closets.

Or--pull the old wall out and build a new, proper one in in its place.

Or, or--buy a closet system that doesn't hang. Get one that sits on the floor.

Maybe what you could also do is at the top drill through the added plywood, and the underlying layers and out the other side and bolt the plywood layers together so that the hanging closet systems also balance each other back to back, and is even less likely to pull forward.

A number of high rise buildings in my area have some partition walls like this. They did not need to be weight bearing, they saved a few inches per unit and they were "fireproof".

Do you think drilling all the way through the wall and bolting the wall braces into both sides thus creating a mirror image of the two closets would work?

nothing at the very top & very bottom of the walls? during the building process they would have had to anchor to something structural?

In the buildings I am thinking of the sandwich is held in place by metal channels anchored to the floor and concrete ceiling.

I would still put up the plywood as well

I'm sure its anchored to the two by four framing in the ceiling (not concrete since this is a single family home) and fastened to the slab too. But it's hard to know for sure without tearing it apart.