] 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.

Before the use of (town) gas for cooking in England you didn't have what we would recognize as a grill or broiler you could cook under and you didn't toast over a fire - try it and you will get something smoky and inedible.

Grilling was done in front of the fire or grate, and toast similarly in the kitchen or nursery as a treat for the children. Americans and others who are familiar with a wide open fireplace for wood, need to image a smaller, more vertical grate with coal and red coals radiating a powerful heat.

The toast was made with a toasting fork, and the cheese softened over a fire, typically with mustard, pepper, sometimes butter, and often porter (ale) or port (wine) and the mixture was spread on the toast, and sometimes, but not usually browned before the fire.

In a sophisticated household there were specially made silver servers with a compartment with hot water to keep the toasted cheese warm to the table, or waiting as a late night snack for the family coming home from the theatre or whatever. More often the toast would be brought to the table with the toasted cheese separate in a saucepan or individual ramekins.

A great treat for the nursery was cinnamon toast - hot buttered toast sprinkled with sugar and dusted with a little finely ground cinnamon - I recommend it.

Stewart