I'm struggling on the application of this. Monitors don't have 960x1080 resolution that I know of, and that's not even close to a standard video resolution.
There are inexpensive video-wall processors which can do something similar to this, but if the signal is originating from a computer, they why not use a video card with two outputs and setup the computer to send half the image to one screen and half to the other.
Still, in 15+ years doing this, the resolutions are you are talking about are extremely strange, and the cost involved may be far more than simply getting standard gear of some sort.
By example: Some displays have built in video wall capabilities. You feed the same image to both displays, and the displays internally split the signal in half and show only the proper half they are setup to display.
So, a bit more information would be helpful as it sounds like you really have two 1920x1080 displays that should spread a 1920x1080 image across both of those displays. You don't do that by making an input 960x1080, you do it by processing the image into two 1920x1080 images, each one containing only half of the original 1920x1080 image. The products to do this, real time, are not inexpensive.
What you are creating is called a 1 by 2 (1x2) video wall.