What is the name for a Type of sandwich containing cheese then grilled?

There isn't one definitive name for a sandwich containing cheese that is then grilled. Here are some common terms depending on the specific type of sandwich and how it's grilled:

General terms:

* Grilled cheese sandwich: This is the most common and widely understood term. It generally refers to a sandwich with cheese and bread that is grilled on a flat surface like a griddle or pan.

* Melt: A more informal term, this can be used for any grilled cheese sandwich or a sandwich with cheese that's melted.

* Toasted cheese: Similar to grilled cheese but often made with a toaster oven or under a broiler.

More specific terms:

* Monte Cristo: This sandwich is made with ham, turkey, and Swiss cheese, dipped in batter, and then deep-fried.

* Croque Monsieur: A French classic, this sandwich is made with ham and Gruyère cheese, grilled in butter, and often topped with béchamel sauce.

* Grilled cheese panini: A panini press is used to grill the sandwich, which gives it a crispy exterior and a melty interior.

Ultimately, the best name for your sandwich depends on the specific ingredients and preparation method.