Preheat the oven to 325 F. Heat 2 tablespoons olive oil over medium-high heat in a large oven-proof pan with lid or Dutch-oven. Season the meats with salt and pepper. Working in batches, add the beef to the pan, without over-crowding, and brown on all sides. Transfer to a bowl and repeat with the remaining beef.
Coarsely chop 2 carrots. Heat 1 tablespoon olive oil in the same pan. Add the chopped carrots, the garlic, and onion. Sauté until slightly softened, about 3 minutes. Add the apple vinegar to the same pan and deglaze over medium-high heat, scraping up any brown bits, and bring to a boil.
Return the beef and any collected juices to the pot. Add the tomato paste to coat the meat and vegetables and stir around for one minute to cook the tomato paste. Sprinkle with the flour and mix to cover the meat and vegetables and to cook the flour. Slowly add the wine while stirring then add the stock, and spice rub so the beef is just covered. Bring to boil, then cover the pot and transfer to the oven. Add thyme bundle and bay leaves. Cook until the meat is very tender, 3-4 hours.
About 30 minutes before the beef is finished cooking, cut the remaining carrots into 1/2-inch slices. Sauté the carrots in a skillet with 1 tablespoon oil until bright in color and crisp-tender, about 3 minutes. Transfer to a plate. Add the onions and mushrooms to the same skillet and sauté until lightly golden and crisp-tender, about 3 minutes. Remove from the heat and set aside.
Remove beef from oven. Pour the stew into a large strainer set over a bowl. Remove the meat from the mixture and return to the Dutch oven, then press down on the cooked vegetables in the strainer to extract as much juice as possible. Discard the solids. Pour the sauce back over beef and add the carrots, onions, and mushrooms. Bring to a simmer to heat through, taste for seasoning and serve hot.