Preheat your grill on MEDIUM-LOW with the stainless griddle or grilling steel in place. Too much heat will burn the bread and undercook the cheese, lower is better.
Thin slice your cheeses and have them ready to place into the sandwich in layers to help the flavors meld. Open the lid and melt a tablespoon of butter on the griddle, spread it around, then place your first two slices of bread on the griddle in the butter. Allow these faces to toast completely in the butter then flip one over and lay the cheese out on the toasted side, place the other piece toasted side down on top of the cheese.
Close the lid, continue toasting, add more butter if necessary, flip, the sandwich can be pressed a little if desired. As the cheese melts out of the sandwich allow it to crisp on the griddle, flip one last time then cut and serve.