In a large ceramic or non stick skillet fry the onions and garlic in a little bit of oil. Once they golden, add the diced tomatoes, grated carrots, basil, and the olive oil. Keep on low heat with the lid on and stir often.
After 15 minutes pour this mixture into a blender or food processor and blend on a low for 5 to 10 seconds, or pulse it a few times. You don't want to liquify it, you just want the carrots and tomatoes to coarsely puree.
Pour this back into the skillet, add the bayleaf, and continue to stir with a wooden spoon, putting the lid on in between stirring, for another 40-45 minutes.
When it starts to get more orange then it's getting closer. ALWAYS stir as often as you can and keep on low. Shut the heat off and cover it when you feel it's rich, subtle, and less acidic. Add the handful of fresh basil once it's cooled.