I discovered this by accident as I was cleaning the fender area by the headlight for painting.
I ran a microfiber cloth, wet with lacquer thinner, across the area I was prepping for paint and I accidentally went over the old, foggy, looking like crap headlight. To my total amazement the opaque layer of the headlight came right out and was left behind was a crystal clear lense !!!
I made a quick pass over the rest of the headlight and I was astounded by the result. A second quick pass, (quick is the trick!), and the results are what you see on the photos!
I immediately did the same on the rest of my cars and 2 years later the headlights still  look this good.
I would recommend trying a small corner of your headlight before doing the whole headlight.

What seems to ensure success is:
1. Use lint free cloth,(I prefer microfiber cloth),
2. Use enough Lacquer thinner,
3. Use a light continuous motion and don't go over the surface again, until the solvent evaporates and the lense "hardens" again  
    (about 30sec).
4. Please, use proper hand protection, ie: Nitrile gloves, when using solvents of any kind. The long term effects of exposing your bare
    skin to solvents is nasty and irreversible!

Give it a try and have a great success with your project!:o)

PS: As an experiment to potentially improve on the original process, about a year ago, I applied a coat of clear lacquer varnish on an old headlight. I sprayed  the coat of varnish on the headlight immediately after I wiped the lense with the lacquer thinner, while the lense was still in the "green" state. A year later, with the headlight seating outside,exposed to the elements, there is no sign of deterioration, pealing or yellowing. I don't think this step is necessary but I add it here as an informative bit. :o)