Heat the oven to 350°F and butter a 3-quart baking dish with 1 teaspoon of the butter.
Melt the remaining butter in a saucepan over medium heat and cook the onion for 5 to 7 minutes, until soft.
Add the soup, milk and process cheese and stir over low heat until the cheese melts into a smooth sauce.
Fold the rice, broccoli, salt and pepper into the sauce until evenly coated.
Spread the mixture in the prepared dish and cover it with foil.
Bake for 20 minutes, uncover and bake for 10 to 15 minutes more, until bubbling at the center, then rest for 5 minutes.