Measure all ingredients and prepare a large, heavy soup pot. Peel the onion, cut it in half and dice it finely. Peel and finely chop the garlic cloves or press them through a garlic press. Wash the basil, shake it dry and pick the leaves from the stems. Set aside a few basil leaves for garnish and roughly chop the remaining leaves. Finely grate the Parmesan and set it aside until serving.

Heat the olive oil in the large pot over medium-high heat. Add the ground beef to the hot pot. Break up the ground beef with a wooden spoon and fry it vigorously for about five to seven minutes. Continue frying until the beef is crumbly and well browned. Lightly season the browned beef with salt and black pepper. Add the diced onion to the beef and cook for two to three minutes until translucent. Add the chopped garlic and cook for about 30 seconds.

Add the tomato paste and cook for about one minute while stirring constantly. Stir in the dried Italian herbs. Pour all of the chopped tomatoes into the pot. Add the beef or vegetable stock and stir everything together thoroughly.

Bring the soup to a strong boil over high heat. Then reduce the heat and cover the pot with a lid. Gently simmer the tomato and beef soup over low to medium heat for about 20 minutes. Stir the soup occasionally while it simmers to prevent it from sticking to the bottom of the pot.

Meanwhile, break the lasagna sheets by hand into bite-sized pieces of roughly equal size. After 20 minutes, carefully add the pieces of lasagna sheets to the simmering soup. Stir the pasta pieces thoroughly so they do not stick together. Simmer the soup for another eight to twelve minutes, until the lasagna pieces are al dente. Stir carefully every one to two minutes while the pasta cooks. If the soup thickens too much, stir in additional hot stock or water a little at a time.

Add about half of the crème fraîche to the soup and stir until smooth. Stir the chopped basil into the soup. Season the lasagna soup with salt, black pepper and, if desired, a pinch of sugar. For a spicier version, add a little chili flakes or cayenne pepper. Remove the pot from the heat and allow the soup to rest for two to three minutes.

Ladle the lasagna soup into deep plates or bowls. Top each serving with a spoonful of the remaining crème fraîche. Garnish the soup with grated Parmesan and the reserved basil leaves. Serve the lasagna soup immediately while hot.