Instead of butter I prefer to use Mayonnaise as the fat spread on the bread to cook in the pan. It does a fantastic job.

this is a good choice because it has less water content than solid butter and promotes better browning. Other options would be to use olive oil, tallow or ghee

I haven't tried actual mayo, but I did once try Miracle Whip (which is my preferred condiment in place of mayo). I would strongly advise avoiding that - the grilled cheese made with Miracle Whip was truly nasty. Proper mayo must be different in a way that makes it more effective in this task.