Preheat oven to 350.
In a large heavy duty sauce pot heat olive oil over medium heat. Add onions, carrots and garlic. Cook vegetables for 10 minutes stirring often to prevent sticking. Add basil, oregano, bay leaves, salt, pepper and tomato paste and cook five to six minutes (This will caramelize the tomato paste resulting in a sweeter sauce). Add diced tomatoes, crushed tomatoes and chicken broth. Reduce heat to low. Allow sauce to cook for 3 1 /2 hours over very low heat, stirring occasionally to make sure sauce is not sticking. Finally, add the vinegar and pulled chicken.
Bring the remaining reserved chicken broth to a boil and cook spaghetti until just tender. Strain spaghetti and add to the tomato sauce. Add Parmesan cheese and stir well. Place mixture in a three-quart Pyrex dish and top with shredded mozzarella.
Cover baking dish first with plastic wrap, then tightly with foil. Bake 40 minutes. Remove from oven and uncover. Let rest for 15 minutes before serving.