I first made one of these about a year ago after I had it in a Florida burger place. It's basically two grilled cheese sandwiches used as the roll. I know, its a ton of calories, but I only have it once a year. The version I made today came out great.
The Bread
Since you are eating 2 sandwiches it's critical to use Arnolds "Very thin" sliced white bread. Each slice is only 40 calories and forms a nice square.
The Cheese
There are lots of wonderful cheeses from around the world to choose from, but I always felt that Velveeta made a great grilled cheese. I know it's a processed cheese like product, but it melts very well, and has a high ooze factor. This cheese choice is no different than having Cheeze Whiz on a Philly cheese steak, which is the preferred choice at Geno's and Pat's.
The Meat
I use an 80% lean freshly ground chuck for my burgers. It has a much beefier taste than ground round or "ground beef". I mix a little fresh ground pepper and salt into the meat and loosely form it into a square shape, so it fits the sandwich.
Condiments
I don't put anything on this burger. I may make a little puddle of ketchup on the plate to dip in.
Directions:
Butter the outsides of the assembled grilled cheeses and brown in medium hot pan. When done remove to a warm plate. Fry the burger till medium rare and place between the sandwiches, putting ketchup, or onion, or tomato, or pickle, inside if desired.
The juices of the burger will drip into the lower sandwich, and the cheese will definitely ooze out the sides. The one I polished off earlier today required 4 napkins. I'm thinking bacon needs to be on next year's sandwich.
