Put all the stock ingredients into a saucepan with 1/2 gallon of water. Bring to a light boil, then lower to a simmer. Simmer for 90 minutes, uncovered. Strain the stock and set aside. Put the chicken into a food storage bag and into the refrigerator.
Heat a tablespoon of olive oil in a heavy, large skillet over medium-high heat. Remove a fistful each of the ground veal and ground pork (put the rest into the refrigerator). Add the meats to the skillet, along with the chicken gizzards and heart. Let the meat brown well, then break it up with a kitchen fork. Remove from the pan with a slotted spoon and set aside.
In a large saucepan or dutch oven over medium heat, heat the remaining olive oil until it shimmers. Add the onions, celery, and garlic until they get soft.
Raise the heat to medium-high. Add the tomato paste and the pureed tomatoes. Cook for about ten minutes, stirring constantly, until it gets thick and noticeably darker in color.
Add the chopped ham and the browned veal and pork. Continue to cook and stir for another three minutes.
Add 5 cups of of the chicken broth, and the thyme, basil, and oregano. Bring to a boil then lower to a simmer. Cover the pot and let it simmer for for two hours. Stir the pot, scraping the bottom well every twenty minutes or so.
After two hours, turn off the heat and let the pot cool for a half-hour. Spoon the contents into a large bowl or food storage container. Put it into the refrigerator overnight to let the flavors come together.
The next day--about two hours before you're ready to serve--put the sauce back onto the stove on low heat. If it seems too thick after it warms up, stir in add a little more chicken stock.
Pull about two cups of chicken meat--a blend of white and dark--from the chicken you used to make the stock. Slice it if necessary into pieces the size of the tip of your little finger.