Start the onions right away so they can caramelize while you’re cooking the burgers. Cut the onions into thick slices, shown below.
Heat olive oil in a medium skillet to medium-low heat; add onions, salt and pepper. Allow cooking slowly, occasionally stirring, until caramelized, about 25-30 minutes.
Next, the burgers: mix hamburger meat, paprika, cayenne pepper, salt and pepper (to taste) and chopped green pepper in a large bowl. Form small sections of meat into small, very thin hamburger-sized patties.
Divide the feta cheese among the patties; try to form the feta into a sort of circle on the top of the patties.
Cover each with another patty and pinch the edges together so that nothing seeps out the sides. Push down slightly in the middle of each patty so they don’t puff up on the grill/skillet (you should make roughly a ¼ inch indent).
Heat grill/grill pan/skillet to medium or medium-high heat; brush with olive oil. Grill/cook burgers to desired ‘doneness’ (I like mine medium-rare); it should take roughly 10 minutes on the first side.
Flip the burgers and finish cooking, about another 10 minutes.
The caramelized onions should also be done.
Serve w/ caramelized onions and lettuce on top of the burger. Enjoy!!