"Toasted cheese" is what in North America is known as a grilled cheese sandwich, and in France usually goes by the name croque monsieur. It's made with toast with cheese on top, then melted, and can be open (i.e. like pizza) or double sided, like a real sandwich. Either way, you can make toasted cheese under a grill (broiler) or in a pan. English toasted cheese doesn't necessarily have ham in it, like croque m. does.

Now, the question is whether this method of "toasting" cheese is the same now as it was in JA's time...I have no idea.

One thing I am sure of though, is that melting un accompanied cheese on a hot surface, which is known in French as raclette doesn't really exist in traditional English cooking.