Melt 1 tablespoon of the butter in a large saucepan over medium heat. Add the onions, carrots, and celery and cook until tender, about 5 minutes. Add three-quarters of the mushrooms and cook until they wilt, 4 to 5 minutes. Add the rice, thyme, and bay leaf and cook for 1 minute. Pour in the sherry or wine and cook until the liquid almost evaporates, 2 minutes. Add 3 cups of the chicken broth along with the beef broth, tomato paste, salt, and pepper. Bring the broth to a boil over high heat, reduce the heat to medium low, and simmer, stirring now and then, until the rice is tender, 30 to 35 minutes.
Remove the thyme and bay leaf; then puree the soup in a blender or food processor (in batches if necessary) until very smooth, 1 to 2 minutes per batch. Return the soup to the pan and stir in the half-and-half over low heat. Keep it warm, adding more chicken broth if necessary to thin out the soup.
Melt the remaining 1 tablespoon of butter in a small skillet. Add the remaining mushrooms and cook until wilted, 3 to 4 minutes. Divide the soup among four bowls and garnish each with the sautéed mushrooms. Serve with the bread for dipping.
Tip: For an even silkier soup, pass the puree through a fine-mesh strainer or food mill.