Is there a "scanning" beam sensor?
The main obstacle to this is that a traditional beam break has a point source and a point sensor. The system as you described would have to have a linear sensor in order to tell you where the break was, which would be more expensive.

It's certainly not impossible. Related work includes "structured light" (combining a scanning or otherwise patterned laser with a camera to give a 3D image), or the Flat Frog system for very large touchscreens. That has a large number of LEDs and photodiodes around the edge and detects fingers by successively illuminating from different angles and looking for the shadow.