Heat oven to 400° F.
Rub the cod with the oil and season with 1/4 teaspoon salt. Dip the top side of the fish pieces (not the bottom side, from which the skin has been removed) in the corn flakes and place on a parchment- or foil-lined baking sheet. Bake until just cooked through and opaque, about 10 minutes.
Meanwhile, steam the broccoli until tender, 4 to 6 minutes. In a small saucepan, over medium heat, melt the butter. Stir in the Parmesan and 1/4 teaspoon salt.
Divide the fish and broccoli among individual plates. Serve with the butter sauce in small ramekins for dipping.