The cheese and beans toastie is the toastie with a controlled flood inside it. A plain cheese toastie is built around a seal: bread buttered on the outside, cheese within, pressed in a hot iron until the edges weld shut and the centre goes molten. Adding baked beans puts a wet, loose, sauced filling inside that seal, and the entire character of the sandwich is decided by whether the seal can contain it. A toastie that holds traps a hot, half-liquid bean-and-cheese centre behind a crisp shell. One that fails splits along an edge and the beans run out down the wrist, which is why this build has a reputation for being good and messy in equal measure.
The craft is heat and load against the seal. The beans are the problem: their tomato sauce is thin and watery, and too generous a spoonful overwhelms the cheese, soaks the inner crumb, and blows the weld before it sets. The fix is restraint and order, a modest layer of beans enclosed on both sides by the cheese, so the cheese melts into a gasket around the beans and does the sealing the beans cannot. The cheese is a strong Cheddar chosen to melt smooth rather than split into oil under the iron. The iron is run hot enough to crisp and weld the buttered outside fast, before the bean liquid has time to migrate through the bread. Pressed correctly, the shell is rigid enough to be cut without the centre escaping.
The sealed-and-pressed frame is where the relatives sit. Ham inside the same iron adds protein instead of sauce and is far less of a moisture problem; an onion or a slice of tomato pushes the water load the other way; the plain cheese toastie is this sandwich with the flood removed. Each deserves its own article rather than being crowded in here.