Toasted cheese is the open relative of the sealed toastie, and the difference is the whole point. There is no second slice and no weld. Bread is toasted on one side, turned, covered in sliced or grated cheese, and put back under the grill so the cheese melts, bubbles, and browns in the open air rather than inside a closed pocket. Because the heat reaches the cheese directly from above instead of conducting through a buttered outer face, the surface blisters and takes on a freckled, lacquered top that a pressed toastie never gets. It is eaten with a knife and fork or in the hand over a plate, a single slice carrying a topping up rather than a filling held in.
The craft is in the order of operations and the position under the grill. The bread is toasted first on the underside so that it has a firm, dry base before any cheese goes near it, because cheese melting straight onto raw bread soaks it into a soft mat that collapses. The cheese is laid right to the edges so the corners do not scorch bare while the centre is still cool, and the slice sits a measured distance from the element: too close and the top burns before the inside flows, too far and the cheese dries to rubber without ever blistering. A strong Cheddar is the usual choice because its sharpness survives the heat and it browns rather than oils out. The result is meant to be eaten the moment it comes out, while the top is still molten and the toast underneath is still crisp.
The variations are a grill-pan tradition of small additions. A base of sliced tomato or a slick of brown sauce or Worcestershire under the cheese is common, the acid cutting the fat. Mustard worked into the cheese sharpens it; a slice of ham beneath turns it toward a meal; Welsh rarebit is the elaborated version, where the cheese is first cooked into a thick ale-and-mustard sauce and then grilled until it bubbles. Each deserves its own article rather than being crowded in here.