Heat the olive oil in a large saute pan over medium-high heat until hot. Reduce the heat to low and add the carrot, celery, onion, and bay leaf. Season with salt and pepper, cover, and cook, stirring occasionally, until the vegetables are very soft but not brown, about 10 minutes. Uncover, add the garlic, and continue to cook for several minutes. Add the tomatoes, parsley, and basil and bring to a very gentle simmer. Cover and cook until the tomatoes have cooked into a puree, about 1 hour. Make sure to stir occasionally so the sauce does not scorch. Season to taste with salt and pepper. Keep hot until needed.

Bring a large pot of water to a boil and add salt. Drop in gnocchi and cook for about 90 seconds from the time they rise to the surface of the water. Place the butter in the bottom of a wide, shallow serving bowl. Warm the bowl and melt the butter, perhaps by placing it over the pot of boiling water.

Drain the gnocchi and place in the prepared serving bowl. Ladle about half of the sauce over the gnocchi, toss gently, and dust with a fine grating of Parmesan cheese. Pass the remaining sauce and cheese at the table.