Cheeses — any of the cheeses that melt well, but some in particular:
cheddar
hard mozzarella
Gouda
...
Sweet, instead of savory:
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.
Use clarified butter only. Don't use regular butter, otherwise it will smoke too much.
The highest heat you should use is medium. It's good to cook it a somewhat low-and-slow.
Try cooking two sandwiches at once, in two separate pans one big pan. Where's why — It really helps to cook each sandwich for 5+ minutes, and 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.
The big downside to this — You end up with twice as many dirty dishes. Two dirty pans, two dirty lids, and two burger presses.
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 (because not moving it means more Maillard), 4) flip and cook (with water+lid) slightly more.
How do I know if it was cooked "done"? The goal is to get enough nice crusty parts on the outside, while also making sure the cheese in the center gets melted.