For a quick fix later on, keep single servings handy: freeze sauce in the sections of a muffin tin. Once frozen, pop them out and store in a plastic bag in the freezer.
In a large saucepan, heat the oil over medium heat. Add the onions and garlic and sauté until translucent, about 10 minutes.
Add the tomatoes, vegetable puree(s), salt, pepper and basil, and simmer uncovered over low heat until the sauce thickens, 30 minutes.
Remove and discard the basil. Taste and season with more salt and pepper, if necessary.