Preheat oven to 350°F. Place garlic on large piece of foil. Pour 2 tablespoons wine over garlic. Wrap foil around garlic to enclose. Roast garlic until soft, about 40 minutes. Cool. Press garlic between fingertips to release from skins; set aside.
Blanch the bacon to remove its smoky taste. Drop bacon slices into boiling water, and simmer 3 minutes. Drain, rinse in cold water, and dry on paper towels.
Cook bacon in large pot over medium heat until brown, about 10 minutes. Transfer to paper towels. Pour drippings into small bowl. Return 2 tablespoons to same pot; reserve remainder. Increase heat to high. Working in batches, add beef to pot and brown, about 7 minutes per batch. (Add some olive oil if necessary.) Using slotted spoon, transfer meat to large bowl.
Reduce heat to medium-low. Add chopped onion and chopped carrot to pot; sauté 5 minutes. Mix in flour. Return beef and accumulated juices to pot. Stir in tomato paste. Add celery rib, herb sprigs, bay leaves, roasted garlic and 2 cups wine; simmer 15 minutes. Add 3 cups broth. Cover; simmer 1 1/2 hours, stirring occasionally. Uncover; simmer until meat is tender, stirring occasionally, about 1 1/2 hours longer. Discard celery rib, herb sprigs and bay leaves.
Cook pearl onions in large saucepan of boiling salted water for 2 minutes. Using 4- to 5-inch-diameter strainer, transfer onions to medium bowl; cool slightly, then peel. Return water to boil. Add carrots and cook until tender, about 4 minutes. Drain. Transfer to bowl of ice water to cool. Drain. (Bacon, stew and vegetables can be prepared 1 day ahead. Cover separately and refrigerate.)
Heat 2 tablespoons reserved bacon drippings in large skillet over medium-high heat. Add mushrooms; sauté until golden brown, about 6 minutes. Add pearl onions to mushrooms; sauté until onions are golden, about 4 minutes. Add carrots; cook until heated through, about 3 minutes. Season with salt and pepper. Bring stew to simmer, thinning with more broth, if desired. Stir bacon and 2/3 of vegetables into stew. Transfer to large bowl. Top stew with remaining vegetables. and some fresh chopped parsley. Serve with crusty bread, noodles, parslied new potatoes, or mashed potatoes.