Heat the oil in a large Dutch oven over medium heat. Add the lardons and cook, stirring occasionally, until golden brown and the fat has rendered, 6 to 8 minutes. Using a slotted spoon, transfer to a paper towel-lined plate and set aside. Reserve the rendered fat in the Dutch oven.
Generously season the short ribs with salt and pepper. In the same Dutch oven, working in batches, brown the short ribs in the remaining bacon fat over high heat, then remove to a plate and set aside. Remove all but 1 tablespoon of fat from the Dutch oven.
To the Dutch oven with the reserved fat, add the carrots and shallots. Cook over medium-high heat until lightly browned, 10 to 15 minutes. Add the tomato paste and cook until slightly darker, 3 to 4 minutes. Add the garlic and cook, stirring, about 1 minute. Remove about half of the carrots and shallots, and reserve.
To the Dutch oven, add the wine, beef broth, cognac, short ribs and bouquet garni. Season with salt and pepper and bring to a simmer over medium-high heat. Cover, transfer to the oven and cook until the beef is very tender, 2 to 3 hours.
Meanwhile, in a skillet over medium-high heat, melt 3 tablespoons of the butter. Add the mushrooms, season with salt and pepper and cook, stirring often, until golden brown, about 5 minutes. Set aside.
Bring the Dutch oven from the oven to the stovetop. Remove the bouquet garni, carrots and shallots and discard. Bring the liquid to a simmer.
Using a fork, in a small bowl, cream together the remaining 1 tablespoon of butter and flour, using your fingers if necessary to mix well. Stir the mixture into the stew. Simmer over medium-low heat for 15 to 20 minutes. Taste and adjust the seasoning as necessary.
Serve the ribs over creamy polenta along with the mushrooms and reserved carrots and shallots. Ladle some of the sauce over the top before serving.
In a medium saucepan over medium-high heat, add the milk, then add the cornmeal gradually, whisking constantly. Add salt and pepper to taste, and bring to a boil, stirring occasionally.
Turn the heat to low, and cook, stirring often, until the polenta is very thick, but is still creamy, 30 to 35 minutes. If the polenta becomes too thick, whisk in additional milk or water, 1 tablespoon at a time, until it is the desired consistency (it should be about the consistency of porridge).
Remove the saucepan from the heat and stir in the butter using a wooden spoon. Taste for seasoning and serve immediately.