Admittedly, not really a sandwich, but I am proud of the thought process I put in this one.
My goal was "cheeseburger quesadilla" on a few minutes notice, with what was present in the house.
First hurdle - no american cheese in the house. And straight cheddar does not actually hit the mark properly in terms of texture or flavor fro a classic burger. However, a mix of shredded cheddar and Gouda does the trick.
Next hurdle - Tomatoes, when cooked, release far too much water to be used in a quesadilla. Substitute jarred roasted red peppers, rinsed, dried, and chopped small but not fine.
Dill pickle, copped just short of fine, will also be called for.
Now, the meat is where technique matters. Some folks (including myself, in the past) make hamburger by making what amounts to a meatloaf mixture. This can yield a really tasty product, but really requires you do a large patty, which isn't what I wanted here. Some folks make a patty out of plain ground beef, and that also is a mistake - they miss the only crucial ingredients, salt and pepper.
So, ground beef, 85% lean, with about 1 tsp of coarse kosher salt (not standard salt, that will be way too much!), and a bit more pepper than you think it should need. Toss to combine thoroughly, but don't make into a patty. Toss into a hot frying pan and crumble, but do not overcrowd the pan - you want maybe a half-pound of meat in a 12" skillet - more than that and the released moisture will stop the browning process that produces much of the flavor - the Maillard reaction doesn't happen in the presence of water. Cook until fully browned.
Now, assemble the quesadilla from the above, and cook as usual. Serve with a ketchup dipping sauce.