Heat oil in a medium size skillet over medium heat.  Add the onion, reduce heat to medium low and cook, stirring, until onion is softened, about 5 minutes.  Cover and cook, stirring occasionally, until onion is a deep golden brown, about 20 more minutes. (instead of the horseradish and cheese, I added mushrooms toward the end of the caramelization...SO great!  I think this would be great with the horseradish but I didn't have any, and LAS is allergic to cheese) 

Shape your ground beef into 8 small patties (about 4 " across), top 4 of the patties with a tablespoon of your onion mixture, top that with another patty and pinch the sides to seal the burger closed.  Repeat and season stuffed burgers on both sides with salt and pepper.  Prepare a grill or grill pan and cook burgers to desired doneness.  About 5 minutes per side for medium.

Ridiculously good! LAS is doing the gluten free thing so he ate his alone, I ate mine open faced style on half an english muffin with ketchup, mustard and a pickle...tres delish!