Puree tomatoes in a food processor. Scrape pulp into bowl, then discard seeds. Set tomatoes aside. Heat oil in a large, heavy-bottomed pot over medium-low heat. Add onions and cook, stirring frequently, until soft, about 20 minutes. Increase heat to medium-high, stir in mushrooms, and cook for 5 minutes. Add veal and pork, breaking up lumps with a wooden spoon and brown for 8 minutes.
Combine tomato paste with 2/3 cup water in a small bowl. Reduce heat to medium, then add tomatoe paste mixture to sauce and cook, stirring for 1 minute. Stir in sugar, nutmeg, basil and tomatoes. Reduce heat to medium-low, season with salt and pepper to taste and gently simmer, uncovered, stirring frequently, for 40 minutes. Adjust seasoning. Makes about 7 cups.
To store sauce, cool to room temperature, then store, covered, in the refrigerator for up to 1 week or frozen for up to a year.