For the soup, heat EVOO, 2 turns of the pan, in a large soup pot over medium to medium-high heat. Add onions, garlic, carrot, salt, pepper, thyme, bay and cook partially covered to soften 10-15 minutes, stirring occasionally. Add tomato paste and stir 2 minutes; add stock, tomatoes and basil. Simmer partially covered for 30 minutes, remove and discard bay leaf.
Purée soup in a blender and return to pot.
To make the béchamel sauce, melt butter in a saucepot over medium heat, whisk in flour and cook 1 minute. Whisk in milk, season with salt and pepper, and let thicken to coat a spoon. Stir béchamel into soup to combine.