To make the meat sauce, in a heavy saucepan over medium heat, warm the olive oil. Add the onion, carrot, and celery and sauté, stirring often, until the vegetables are soft, about 10 minutes. Stir in the parsley and, after 1 minutes, add the meat, breaking it up in the pan with a wooden spoon. When the meat loses its pinkness, after 5 to 7 minutes, raise the heat to high, add the wine to the pan, and stir until the alcohol has evaporated and the liquid has reduced, about 3 minutes. Reduce the heat to medium, add the tomatoes, season with salt and pepper, and cook, stirring frequently, until the liquid has reduced slightly, about 10 minutes.

Add 1/4 cup of the broth, reduce heat to medium-low, cover partially, and simmer, stirring from time to time and adding additional broth if the sauce is too dry, until the sauce is thick and the flavors are blended, about 40 minutes.

Bring a large pot 3/4 full of water to a boil. Add the pasta and the coarse salt, stir well, and cook until tender but still quite firm to the bite, about 7 minutes.

Transfer the sauce to a large frying pan placed over medium0high heat. Add the cooked pasta and the butter and stir until the sauce is amost completely absorbed, about 7 minutes.

Transfer to a warmed serving bowl and serve immediately. Pass the Parmesan cheese at the table.