Melt the butter in a 3-quart saucepan over medium heat. Add the onion and cook, stirring often, until softened but not browned, about 6 minutes. Add the stock, mushrooms, rice, and salt and bring to a boil. Reduce the heat, cover, and simmer about 30 minutes.

Pull the thick center vein from the sorrel leaves. Put half of them in a blender along with half of the chives, pour half of the boiling-hot soup over them. Hold the lid down with a dry towel and turn on the blender on low speed, then gradually increase to high. Blend until very smooth. Pour into a second saucepan and repeat with remaining sorrel, chives, and soup. Stir over medium heat until it almost simmers; don’t let it boil.

Ladle into warm bowls, place a small mound of sorrel strips in the middle of each serving. Stir the créme fraîche with a spoon and thin with cream until it falls off the spoon in a thick stream. Hold a spoonful about 6" above each bowl full of soup, and let it fall in a circular or zigzag pattern.