Not exactly roofing

Here's my problem. I recently redid my office and the last thing I have to do is mount a flat screen on the wall. I bought a 37" tv (I didn't want it too big to hang on a swivelling, full motion mount). When I tried to find a stud, I came up empty. Even when the stud detector said one was there. So I removed an outlet cover and discovered that the builder had wrapped the walls with 1 1/2" rigid insulation under the sheetrock.

Now, I think the wieght on the tv bracket would eventually cause it to start digging into the sheetrock and/or insulation. The bracket is 3 1/2 " x 12". So I thought about cutting a piece of 2x4 the size of the bracket, cutting a hole in the sheetrock and insulation that size and putting up the 2x4 as backing. But that would make 5 bolt/screw holes (3 for the tv bracket and 2 to attach the 2x4) in the middle of a relatively short 2x4, and likely to split. So now I'm pondering making backing out of two pieces plywood screwed together to the desired thickness and mounting to that. Anybody see any problems with that or other solutions?

I agree with your engineering effort. Dimensional lumber probably would split if you were anchoring to it.

The alternative would be to drill "through" holes through the "filler" dimensional lumber (if you used it) and anchor into the existing stud. The "filler" would then be acting more like a shim.

I'd use the plywood.