Some thicker preserves, the less moisture the better. Something really tart like cranberry preserves could provide an acid that pairs very well with the fatty cheese.
Sources of acid (very important, see description below):
pickles I personally love when the pickles are heated on the bare pan for a minute, before they're put inside the sandwich.
Ignore Adam's cooking method, and start with steaming the two sandwich halves, open-faced. This encourages more cheese to squeeze out, which improves the fried-cheese-against-pan effect that happens near the end of the process.
Be sure to use a high-heat oil like clarified butter. Some of the oil can end up pooling at one side of the pan, and getting really hot over time.
Burner heat level — You want to go slow-and-slow. It should take ~6 minutes to cook a sandwich.
When cooking one sandwich, use medium heat. For two sandwiches at once, ⅔ max heat is good.
Try cooking two sandwiches at once. Here's why — it's good to take 5+ minutes to cook a sandwich, but then I feel pressured and rushed if Tiffany is sitting there with her hot sandwich in front of her, waiting while I'm hurrying to finish mine.
Prepare both sandwiches first, so you can start cooking them at the same time.
You can do this in two separate pans, but you end up with twice the number of pans to wash, and twice the number of lids.
I find it better to use one large pan, and use the "water in a pot, to serve as a burger press" trick. Since you're using a pot, you can find one that's fairly big.
The phases I go through — 1) cook on one side with the weight, 2) cook on the other side with the weight, and push down a fair bit, 3) remove the weight, add water and cover, cooking the same side.
Doneness level:
There are two main goals here: 1) Get the cheese in the center of the bun thoroughly melted, and 2) make the outside of the bun crispy enough. Goal #1 is the harder of the two, since you can't know if it's complete until you cut into it.
It's better to err on the side of over-done, to be sure the cheese gets completely melted. The only real downside to over-cooking it is that the outside could be a little too darkened, but that's relatively easy to control. Usually there aren't any other downsides for cooing it longer than needed.
Balancing the flavors:
There's a LOT of fat on these, with all the cheese. It can help to add a lot of acid to balance that. Options that work great for this — pickles, TODO.
dishes where the cheese stuffing is a popular variant, but that the stuffing is optional (and the cheese variant is possibly less popular than the non-cheese version)
dishes that are bread and cheese integrated together somewhat, but they don't have a real solid chunk of melted cheese in the middle — Cheese bread, pan de queso