Heat the oil and butter together in a pot. Add the chopped onion to them and fry for about 5 min, stirring often, until the onion softens. Remove it from the heat before it turns brown in color.
Once it comes to a boil, lower the heat and let simmer for about 20 min. Cover half the pot with a lid so that you can stir from time to time. Add the basil and blend the soup. Then filter it into a clean pot.
Add the cream and put it to heat but be careful not to have it start boiling. If necessary, add a little more broth. Then remove it from the heat and distribute into portions. Serve thus prepared soup garnished with basil leaves.