Stir the broccoli florets, cream of chicken soup, and 1 tablespoon softened butter together in a 9x13-baking dish until the broccoli is coated with the soup. Pack lightly to form a flat surface. Layer about half the slices of processed cheese food atop the broccoli mixture.
Bake in the preheated oven until the cheese is bubbling, about 20 minutes.
Remove the dish from the oven; stir to blend the melted cheese into the broccoli mixture. Top with the remaining processed cheese food. Top with the crushed buttery round crackers.
Return dish to oven and bake until the added cheese begins to melt, about 15 minutes more. Allow the casserole to cool and thicken slightly before serving, at least 15 minutes.