Heat oven to 300° F.
Season the meat on all sides with 1 tablespoon of the salt and the pepper.
Heat the oil in a Dutch oven or large ovenproof pot over medium heat. Cook the meat in batches, turning to brown all sides. Transfer to a plate.
Add the wine to the drippings in the pot and bring to a boil. Cook for 3 minutes, using a wooden spoon to scrape the bottom of the pot. Add the meat, tomatoes, and 2 cups of the broth and bring to a simmer. Add the yellow onion, whole carrot, and garlic. Cover tightly and transfer to oven for 1 hour.
Add the baby-cut carrots or carrot strips, cover, and return to oven until the meat is fork-tender but not falling apart, about 30 minutes more. Skim any fat from the surface.
Meanwhile, melt 1 tablespoon of the butter in a large skillet over medium heat. Add the pearl onions and the remaining broth. Simmer until the liquid almost completely evaporates and the onions are tender, about 15 minutes, shaking the pan occasionally to coat the onions with the liquid. Transfer to a plate.
Clean the skillet, then melt the remaining butter over medium heat. Add the mushrooms and toss to coat. Cook, stirring occasionally, until the liquid released by the mushrooms almost completely evaporates, about 12 minutes. Remove from heat and add the remaining salt.
Meanwhile, place the pot of beef stew on the stovetop over medium-high heat. Using a slotted spoon, remove and discard the onion half and the whole carrot. Simmer the stew, uncovered, until it thickens to the desired consistency, 10 to 20 minutes. Add the pearl onions and mushrooms and simmer for 5 more minutes. Spoon onto plates.
In Advance: Beef Bourguignonne's flavor improves with time. Make it 1 or 2 days ahead of time, then cover and refrigerate it, keeping the onions and mushrooms separate from the stew. The next day, skim any fat from the surface. Reheat the stew over a medium-low flame for 45 minutes.