1. Bring broth to a simmer in a large saucepan over medium-high heat. Meanwhile, remove stems from mushrooms. Add thyme sprigs, mushroom stems, and half of parsley to broth. Reduce heat to low; gently simmer until ready to use. Remove leaves from remaining parsley, and coarsely chop. Discard parsley stems.
2. Sauté livers in hot oil in a Dutch oven over medium heat 5 minutes or until cooked. Remove livers, and finely chop. Add 4 Tbsp. butter and mushroom caps to Dutch oven, and sauté 10 minutes or until deep golden brown. Remove from Dutch oven using a slotted spoon. Add onion, and sauté 10 minutes or until tender. Add celery, and sauté 5 minutes or until softened.
3. Add rice, and cook, stirring constantly, 1 minute or until fragrant. Stir in wine, and cook, stirring often, 3 minutes. Add 1 cup hot broth mixture (discarding mushroom stems and herb sprigs), and cook, stirring constantly, until liquid is absorbed. Repeat procedure with remaining broth mixture, 1 cup at a time, until liquid is absorbed. (Total cooking time is 30 to 35 minutes.) Remove from heat.
4. Stir in livers, mushroom caps, cheese, chopped thyme, chopped parsley, remaining 2 Tbsp. butter, and salt and pepper to taste. Serve immediately.