In a large, deep skillet, heat the butter over medium-high heat until foaming. Add the olives and cook, stirring, until sizzling, about 2 minutes. Add the tomato sauce and bring to a simmer. Remove from heat and set aside.

Bring the salted water to a boil and then add gnocchi. Cook gnocchi, stirring gently until tender, about 1 minute after they rise to the surface of the pot. Drain the gnocchi.

Bring the tomato sauce back to a simmer over low heat. Add the gnocchi and stir gently with a wooden spoon until coated. Stir in the grated cheese and basil. Season with salt and pepper to taste. Serve immediately.