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.
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. 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 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 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.