Prepare the vegetables: Thinly slice the mushrooms; press or mince the garlic; discard the dark green part of the leeks, keep the white and light green parts, then slice them thin.
Melt the butter in a pot or saucepan over medium-low heat. Add the leeks and garlic, then sweat about 10 min until they are softened.
During this time, heat the oil in a separate pan over medium-high heat. Add the mushrooms and sauté them in batches, so that they will turn golden-brown rather than 'boil' in the water released during cooking. Take the mushrooms out of the pan as they become golden-brown and set aside.
When the leeks are softened, stir in the flour. Cook 2 min, with stirring, then add the rice and mushrooms (you may keep some mushrooms aside as a garnish). Pour in the broth, bring to a boil, then lower the heat and simmer about 15 min, until the rice is cooked. Season with salt and pepper to taste.
Purée the soup in a blender, then distribute it into bowls. Garnish each bowl with a spoonful of sour cream, the optional chopped chives, and reserved mushrooms. Serve.