Preheat the oven to 350 degrees F.
Bring 6 quarts water to a rolling boil and add 2 tablespoons salt. Set up an ice bath nearby.
Using a vegetable peeler, carefully peel any of the tough string from the celery, removing almost an entire outer layer. Then, cut the celery into 2-inch pieces. Add the celery to the boiling water and cook until tender, about 10 to15 minutes. Refresh the celery in the ice bath then set out to air dry.
Roughly chop the chicken livers. In a small bowl, combine the chicken livers and ground veal. Add the olive oil and mix well until all the ingredients are combined. Add the salt and pepper, to taste. Add the red wine, pecorino, parsley and mix well.
Using your fingers, gently stuff the filling into the celery "shells". Place in a lightly oiled baking pan. Ladle the tomato sauce over the stuffed celery and bake until hot and bubbly, about 30 minutes.
In a 3-quart saucepan, heat the olive oil over medium heat. Add the onion and garlic and cook until soft and light golden brown, about 8 to 10 minutes. Add the thyme and carrot and cook 5 minutes more, until the carrot is quite soft. Add the tomatoes and juice and bring to a boil, stirring often. Lower the heat and simmer for 30 minutes until as thick as hot cereal. Season with salt and serve. This sauce holds 1 week in the refrigerator or up to 6 months in the freezer.