In a large stock pot with cover, saute the onions and leek in the butter, about 5 minutes. Add the diced green beans, diced red potatoes, canned tomatoes and beef stock. Cover and simmer, gently, for 30 minutes.
After the 30 minutes, add salt and pepper and broken spaghetti. Cover and cook for 15 minutes longer.
During the 15 minutes while the spaghetti is cooking, prepare the tomato paste mixture. In a bowl, mash the garlic. Add the basil, thyme and sage to the garlic. In a separate bowl, add egg yolks and lightly beat. Gradually add the olive oil, tomato paste and cheese to the egg mixture. Combine the garlic and seasonings to the tomato paste mixture.
When the 15 minutes are up from step # 2, add just a little soup (1-2 ladles)into the tomato paste mixture. Stir completely. You will now gradually stir the tomato paste mixture, into your large stock pot. Stir, cover and simmer another 5 minutes.
Serve with your favorite Italian bread and you may want to add more cheese on top of the soup. Delicious.