For the purest, tastiest tomato sauce, it's a good idea to remove the skin and seeds first from the tomatoes. To remove the skins, score both ends of the tomatoes with a sharp knife by making a small "X."
Place the tomatoes in boiling water for 2-3 minutes. Remove and submerge in an ice water bath. Let rest for 1 minute.
Take the tomatoes out of the ice water and peel the skins off. They should just push off with your thumbs. Cut the tomato in half and scoop out most of the seeds (save for a soup or juice.) Chop tomatoes roughly in preparation for the sauce.
In a heavy-bottomed pan heat the olive oil over medium low. Add the onions, carrot, basil and garlic. Cover and cook for about 20 minutes. Remove the lid and stir often. Lower the heat if vegetables are starting to color.
While the vegetables are softening, prepare the tomatoes (skip this step if you already blanched, skinned and seeded them.) Slice the Romas in half vertically, from stem to base. With your thumb, take out the pulp. Don’t worry if a few seeds remain. (You can also strain the pulp to make a glass of wonderfully fresh tomato juice.)
Add the tomatoes to the sauce pan and raise the heat to medium. Stir the vegetables and the tomatoes together. Cook the sauce just under the boil. It is important to evaporate much of the moisture.
When all the ingredients have cooked and become very soft, run the mixture through a food mill or blend and push it through a fine-mesh screen. If using a screen, cook the sauce a little longer to completely break down all the ingredients.
After passing the sauce through the mill or the screen, return it to the pan and place over low heat. Add a little salt and pepper. Stir the sauce occasionally. At this point all you are doing is evaporating liquid and thickening the sauce.
When the sauce has become thick remove it from the stove. Set aside. For pizza, a thicker sauce is best.