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.
Pure soup in a blender and return to pot.
To make the bchamel 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 bchamel into soup to combine.
For the noodles, beat eggs with cheese, salt, pepper and parsley. Melt butter in a 12-inch skillet over medium heat and add half of the egg mixture. Swirl into a very thin, large layer, cover with loose foil and cook to completely set, 3-4 minutes. Repeat with remaining egg mixture. Remove pan from heat and roll up eggs into 2 logs. Slice each log into 1/4-inch noodles.
Serve soup in shallow bowls topped with a few egg noodles.