In a large soup, heat the olive oil over medium high heat. Add the onions and cook, stirring occasionally until the onions are soft, 8 minutes.
With kitchen string, tie the basil sprigs together. Add basil sprigs, tomato puree, and chicken stock to the onions and bring the soup to a boil. Reduce heat to medium-low and simmer until the soup slightly thickens, about 10 minutes. Allow it to cool for 10 minutes.
Remove the basil and discard. Season the soup with salt and serve it warm with a garnish of basil strips.
The soup can be done up to one day in advance and stored in the refrigerator until ready to serve. If so, heat the soup over medium heat until hot, for 5 minutes.
Ladle the soup into bowls and garnish with basil strips.