Wipe a large soup pot with the oil until it is covered in a thin film, pour in the rest of the oil, and heat the pot over medium-high heat until the oil shimmers. Sauté the onion until soft, about 6 minutes. Add the green garlic and continue cooking until it turns fragrant, another minute or two.
Add the rice, stock, parsley leaves, and bouquet garni; bring them all to an irritated boil. Reduce the heat to low; then add the lettuce and salt. Partially cover and simmer for 30 minutes. Check the rice periodically for doneness, especially if you are using brown or wild-rice blends.
Remove and discard the bouquet garni. Blend the soup in batches until it is smooth and return to the pot with lots of pepper. Heat for 1 minute over medium-low heat to meld the flavors.
At this point, the soup can either be served immediately or go into the fridge for reheating the next day (if the latter, add a little broth first to soften it).
Garnish each bowl with the additional fresh herbs and Parmesan.