I call this a "knife-and-fork-job" because it marries the juiciness of a French dip sandwich with the casseroles of my youth. The best part of eating a good French dip sandwich is that moment when the bread has taken on all the juice it can without disintegration—sort of the ephemeral cherry blossom of sliced beef, the green flash right as the sun dips below the ocean horizon. Brennan & Carr in Brooklyn creates this French dip nirvana routinely, and it is their sandwiches that inspired this dish. If you ever get to Brennan & Carr, make sure to order the fries and root beer in a frosted mug!
In a large pan set over medium-low heat, heat the olive oil for about 1 minute. Add the onions and cook, stirring infrequently, until they are brown and caramelized, about 35 minutes. Remove the pan from the heat and set aside.
In the same pan over high heat, combine the beef broth and the onion soup mix. Bring to a boil, then reduce the heat to a simmer. Add the slices of roast beef to the pan. Simmer the strips for about 10 minutes. Remove the roast beef slices to a bowl, reserving the broth.
Preheat the oven to 350°F. Grease the 9-by-13-inch casserole dish with 1 tablespoon of the melted butter. Slice the top halves of the rolls off and set aside. Place the bottom halves of the rolls in the casserole dish. Top with the caramelized onion and roast beef slices. Fill the pan with just enough beef broth to cover the beef. Top with cheese slices and then the top halves of the rolls. Brush with the remaining tablespoon of butter and cover
Bake the casserole for 25 minutes. Remove the foil and bake on an upper rack until the roll tops are golden brown, another 5 minutes.
Cut into individual sandwiches and serve hot.