Heat 1 tablespoon of the olive oil in a large, heavy saucepan until shimmering. Add the onion, carrot, celery and pancetta and cook over moderate heat, stirring occasionally, until the vegetables are softened but not browned, about 8 minutes. Scrape the vegetable mixture into a large bowl.
Add the remaining 3 tablespoons of olive oil to the saucepan and heat until just shimmering. Add the beef, veal and pork and cook over moderately high heat until just barely pink, about 5 minutes. Return the vegetable mixture to the saucepan. Add the garlic and cook over high heat until fragrant, about 1 minute. Add the wine and cook, stirring occasionally, until almost evaporated, about 8 minutes. Stir in the tomatoes and their juices, the chicken stock, thyme and bay leaf. Season with a generous pinch of salt and pepper and bring to a boil over high heat. Cover partially and cook over moderately low heat for 1 hour. Discard the bay leaf. Stir in the heavy cream and cook the sauce just until heated through.
In a large pot of boiling salted water, cook the pasta until al dente. Drain well, return to the pot and toss with the sauce. Serve the pasta in deep bowls and pass the Parmesan at the table.
Make Ahead: The sauce can be refrigerated for up to 3 days and frozen up to 1 month.
Wine Recommendation: A Barbera from Italy or California has enough weight and flavor to stand up to the rich meat sauce and enough acidity to balance the tomatoes. Consider the 1997 Louis M. Martini from California or the more complex 1996 Barbera d'Alba Costamiole Alfredo Prunotto from Italy.