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Preheat the oven to 450 degrees F.
Cut the bacon into 1/4-inch cubes. Saute the bacon in 4 tablespoons of the olive oil in a braising pan over medium heat until brown, then remove from the pan using a slotted spoon.
Sprinkle the flour over the meat to coat evenly. Place the braising pan in the oven for 5 minutes. Remove the pan, toss the meat, and return it to oven until the meat is browned, about 5 more minutes.
Remove pan from oven and reduce the oven temperate to 300 degrees F.
In a separate saute pan, saute the onion and shallot with a pinch of salt and pepper over medium heat until soft, about 4 minutes. Add the bacon.
Heat the braising pan on the stovetop over medium heat. Add the remaining 3 cups of red wine, 3 cups of the beef stock, the tomato paste, minced garlic, cooked shallots and onions and the herb bouquet. Bring to a simmer, cover, and place the braising pan in the oven. Cook until the meat is tender, about 3 1/2 hours.
When there is about 1 hour left for the meat to cook, heat the duck fat in a medium pot to 200 degrees, and then add the smashed garlic, the remaining 2 sprigs thyme and some salt and pepper. Add the potatoes and cook, keeping the fat at 200 degrees, until soft, about 45 minutes.
In a separate saute pan, saute the pearl onions in the butter over medium heat until brown. Add the remaining 1/2 cup beef stock and some salt and pepper and simmer until tender, about 30 minutes.
In a separate saute pan, saute the mushrooms in 1 tablespoon olive oil over medium-high heat until tender, then season with some salt and pepper.
When the meat is done, strain the liquid from the braising pan into a saucepan. Discard the herb bouquet and reserve the meat.
Return the meat to the braising pan and top with the pearl onions and mushrooms.
Begin to reduce the liquid in the saucepan over high heat; adjust the thickness of the sauce by continuing to reduce if too thin or adding stock if too thick.
Pour the sauce over the meat and vegetables in the braising pan and bring to a simmer. Serve the bourguignon in a bowl surrounded by the duck fat-poached potatoes.