Preheat the oven to 350 degrees F.
Heat 2 crepe pans over medium heat, smear with oil, ladle in 2 ounces of batter and cook the crespelle, 2 at a time until all batter is used.
In a 12-inch saute pan over a medium-high flame, heat the olive oil until hot but not smoking. Add the onion, carrot and celery, and cook until soft and golden brown, about 5 minutes. Add the chicken meat, livers, and nutmeg. Season, to taste, with salt and pepper. Lower the heat to medium and let cook until the meat is browned, about 15 minutes. Remove from the heat and set aside to cool at room temperature.
When the meat mixture has cooled, add the eggs and 1/2 cup of Parmigiano. Use a wooden spoon to mix until well combined. Scoop 1 1/2 tablespoons of the meat mixture into the center of each crespella. Roll the crespella into a tube along the long side and place in a lightly oiled baking dish, in a single layer, seal side down. Repeat with the remaining cannelloni, placing them side by side in the baking dish. Ladle the tomato sauce over the cannelloni. Sprinkle the crespelle with the remaining Parmigiano and bake in the oven until the cannelloni are hot throughout and the cheese is hot and bubbling, about 30 minutes. Serve immediately on warmed plates.
Place the flour in a mixing bowl, add the eggs, and whisk them in. Add the salt, then whisk in the milk, a little at a time until all the milk is incorporated. Allow the batter to stand for 20 minutes.
Heat a 6-inch nonstick skillet over high heat until hot and brush with olive oil. Turn the heat down to medium and pour 1 1/2 tablespoons of butter into the pan. Cook until pale gold on the bottom, about 1 minute. Flip and cook just 5 or 10 seconds on the second side. Remove and set aside. Continue the process until all the batter has been used.
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.