Rinse and dry beef in paper towels; it will not brown if it is damp. Heat 1 to 2 tablespoons of olive oil in a large frying pan until almost smoking (add more as needed). Add beef, a few pieces at a time, and sauté until nicely browned on all sides. When done, add to the Slow Cooker.
Add 2 ½ cups wine and 2 to 3 cups stock to the Slow Cooker, just enough so that the meat is barely covered.
In the same large frying pan, deglaze the pan with ½ cup of the wine, and then add the liquid to the Slow Cooker. Add 1 to 2 tablespoons of olive oil to the pan. When hot, sauté the sliced yellow onion, adding the garlic when onions are about half-way done (onions should begin to caramelize and brown, but be careful not to burn the garlic.)
Add the carrots to the Slow Cooker.
Add the onions and garlic, tomato paste, bouquet garni, and ½ teaspoon salt and ¼ teaspoon pepper to the Slow Cooker.
Once the contents of the Slow Cooker come to a low boil, set the Slow Cooker to low. Regulate the heat so that the liquid simmers very slowly for at least 3 to 4 hours. The meat is done when a fork pierces it and it begins to pull apart easily.
Wipe out the frying pan and heat remaining oil and butter over high heat. As soon as you see butter has begun to subside, indicating it is hot enough, add mushrooms. Sauté then for about 4 to 5 minutes. As soon as they have begun to lightly brown, add them to the Slow Cooker.
About ½ hour before the beef is ready, start cooking the peeled pearl onions. Heat 1½ tablespoons butter with 1½ tablespoons of olive oil until bubbling in the frying pan. Add onions and sauté over moderate heat for about 10 minutes, rolling them so they will brown as evenly as possible. Be careful and try not to break their skins. You cannot expect them to brown uniformly.
Add ½ cup of the stock to the frying pan. Cover and simmer slowly until the onions are perfectly tender but hold their shape, and most of the liquid has evaporated. Add onions and any remaining liquid to the Slow Cooker.
Once the beef is tender, remove the bouquet garni.
Transfer the Boeuf a la Bourguignon to a nice serving bowl, or arrange it on a platter surrounded by potatoes, extra-wide noodles, or rice, and decorated with a few sprigs of Italian parsley.