Ed, CaterChef's recipe is similar to some modern variations called Hot Browns or KY Hot Browns (less the turkey, of course). But it doesn't come close to the actual dish.
Hot browns are made with Mornay sauce, not cheddar. And never contained ham, as some current versions do. When it originated at Brown's Hotel, in Louisville, in the '20s, poached chicken was the protein of choice. Later on turkey was substituted, and the two are considered acceptible alternatives.
I've never been able to uncover when and where tomato started to be used as a garnish. It's so common, however, that it's done that way at Brown's today. The original used mushrooms, though. Given the timeframe, these were probably carved mushrooms, but the record isn't clear.
A signature of Hot Browns is how the bread is laid out. It consists of two slices, one of which is cut on the diagonal. These triangles are fitted against the center square. Poached turkey or chicken is laid out over this elongated triangle, Mornay poured over that. The whole thing is topped with additional Parmesan, a pair of crossed bacon strips, and the mushrooms (tomatoes, nowadays), and popped under the broiler until hot and bubbly.
As a matter of historical interest, Brown's is known for something else even more important. During WW II, colored military personel, in uniform, were welcomed at the bar and in the dining room at Brown's---something unheard of anywhere else in the South at the time.