1. Heat oil in a large saucepan over medium heat. Add onion and garlic and cook until softened, about 5 minutes. Add tomatoes, break up coarsely with a potato masher, and bring to a simmer. Reduce heat to low and simmer for 1 hour.
2. If tomatoes are still chunky, mash again. Add cheese rind, diced meats (if using) and oregano and cook until very thick, about 30 minutes more. Season sauce with salt and pepper. Remove from heat and let cool to room temperature.
3. Transfer sauce, including Parmesan rind, to an airtight container and refrigerate for up to 1 week or freeze for up to 3 months. (Remove cheese rind and garnish with basil leaves before serving.)