Cut thin slice off top of tomatoes. Peel tomatoes, using either serrated swivel-blade vegetable peeler or hot water method. For this method, drop 2-3 tomatoes at a time into large pot of boiling water until their skins crack, 1-2 minutes. Immediately transfer tomatoes to bowl of ice water. When tomatoes are cool enough to handle, use your fingers to pull off skin. Halve tomatoes lengthwise and use your thumb to push out seeds, then your fingers to remove pulpy ribs. Chop tomatoes and set aside; there will be 6-7 cups.
In large, heavy pot, heat oil over medium-high heat. Add onion and cook for 3 minutes, stirring occasionally. Add garlic and cook, stirring often, until onion starts to color, 3-4 minutes. Add tomatoes and oregano and stir well. Cook, uncovered, for 15 minutes. Taste sauce, adding sugar if it is too acidic. Mix in basil and cook until tomatoes have broken down to your taste, 10-15 minutes for chunky sauce, 12-15 minutes for pulpier one.
Note: Using ripe tomatoes is important. Supermarket ones usually require sitting at room temperature for 5 to 10 days to turn really red. They make a more chunky and drier sauce than local tomatoes in season because they are less juicy.