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Taylor Ham can be done up all sorts of ways - in omelettes, in breakfast sandwiches, in mac n cheese, etc - but this is the best way to experience it for what it is.
This recipe is straight from my childhood memories of my mom's favorite townie burger joint: Anthony Wayne's, on 46. Now long gone. They cooked all their stuff over charcoal - I can't pull that off often, but for added authenticity, do it over fire.
If your Taylor Ham is unsliced, you'll want to slice 4 thin (1/8") slices off the end; if sliced, that's half a package, ie 4 slices of thin-sliced or 2 slices of thick. It's a preference thing but I like thin better, more surface area = more char = more delicious.
Preheat a pan til just before it smokes. Add the ham to the pan, avoiding the center of the pan where it is naturally hotter. As the ham cooks it will contract, raising the center off the pan and only cooking the edges, so you will either want to notch the slices with the edge of a spatula so they don't bubble up, or press them into the pan with a weight.
Fry the slices of ham til slightly browned, then flip them and, if necessary, press them into the pan again to cook the other side.
Layer the ham in an overlapping spiral staircase pattern so that the borders of the stack roughly match the size of your bread or roll (I hugely recommend potato rolls in general, but especially for this). Cut a slice of red onion and put it directly onto the pan, the put the stacked ham on top of it so the ham is off the pan and the onion starts to cook.
Toast your bread, either in pan with the ham and onions (butter it first if you want) or in a toaster.
Once the onions start to sizzle, top the onion and ham stack with a slice or two of American cheese. Let the cheese soften around the edges; once the bread is slightly crispy (or done toasting), put it on a plate, and lid the pan so the cheese melts. If the onions are over-cooking or at risk of burning, pull the pan off the heat and let the cheese melt without direct heat.
Spatula the stack onto the roll. Add hot sauce or ketchup if you like.
There you go. Lunch, Jersey style.