Heat the oven to 350°F (176°C). Roll the beef cubes in flour. Heat 4 tablespoons each butter and olive oil in a large skillet over high heat and brown the beef, in batches, on all sides.
Return all the meat to the skillet. Sprinkle the meat with salt and pepper, pour the Cognac over it, and carefully ignite. When the flame dies, put the meat in a casserole.
Add the bacon, garlic, carrots, leeks, onions, and parsley to the skillet. Cook, stirring, until the bacon is crisp and the vegetables are lightly browned.
Transfer the bacon and vegetables to the casserole with the meat and add the bay leaf, thyme, Burgundy, and enough water barely to cover the meat. Cover and bake for 1 1/2 hours.
Prepare a beurre manié by blending 1 tablespoon each butter and flour. Stir into the casserole bit by bit. Return the casserole to the oven and continue cooking until the meat is very tender, 30 to 60 minutes longer.
Meanwhile, brown the small onions in 2 tablespoons butter with a dash of sugar in a large skillet. Add a little water, cover, and cook until the onions are almost tender. (If using frozen onions, follow the cooking directions on the package, then brown them in butter with a little sugar.) Set aside.
Sauté the mushrooms in the remaining 2 tablespoons each butter and oil in a large skillet until lightly browned on one side. Sprinkle them with lemon juice and turn to brown the other side.
To serve, add the onions to the casserole and garnish with the mushrooms and parsley.