According to the Sterns in Blue Plate Specials, most places have a steamer box, where the burger and cheese are cooked separately, and the glob of cheese is then added.
Here is there advice to make it at home:
You don’t have to be a do-it-yourself type to rig your own steam box at home. Using Brian O’Rourke’s expert advice, we jury-rigged one using a large skillet, a metal colander with feet, a big heatproof bowl, and a clean, empty 6-1/2-ounce tuna can.
Here’s how: Put about one inch of water in the skillet and place on the stove over medium heat. Place the colander above the water. Put 4—6 ounces of cheddar cheese in the tuna can and, when the water begins to boil, put the tuna can in the colander. Then, cover the colander with the big heatproof bowl, trapping the steam inside. In only a matter of minutes, the cheese will melt, then all you have to do is use a spoon to scoop it out onto a hamburger, a hard roll, or a piece of apple pie.
Makes enough cheese for 3—4 servings.
P.S. It is actually possible to cook the hamburger in the makeshft steam box if you want to be a classicist about your steamer. Put a half-inch patty of lean raw ground beef in another tuna can and steam both burger and cheese at the same time. http://www.roadfood.com/Reviews/Overview.aspx?RefID=118