Heat oil in a large heavy bottomed skillet over medium-low flame. Add onions and jalapeños and cook, stirring occasionally, for 25 minutes. Add sugar and cook for another 5 minutes, then remove from pan. Top burgers with a few tablespoons of the onion/pepper mixture before serving.
For the burgers:
Place a large heavy skillet over medium-high heat (or prepare your grill as needed).
In a small bowl mix together the cheeses.
Place ground beef in a large bowl. Add in brown sugar, hot sauce, worcestershire, garlic, salt and pepper, mix well with hands.
Divide meat into 4 patties, then remove 1/4 of each patty, rolling the meat into a small ball. Reform the remaining meat into deep dish patties (like the image shown in the post), being sure there's a cavity in the center of each patty. Fill each cavity with 1/4 of the cheese mix. Flatten the remaining small balls of meat into small disk shapes, then use them to seal the tops of the burgers, covering all of the cheese, and pinching the edges shut as best you can. You shouldn't be able to see any of the cheese.
Place one tablespoon of butter in the skillet, add two patties, and cook on each side for 3-4 minutes, depending on how well you want the burger cooked (3 minutes will be medium-rare, 4-5 for medium, and 6 for a well done burger). Place burgers on buns, and top with caramelized onions and peppers. Serve at once!
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