Once the beef is tender, carefully remove all solids from the stew sauce (except the beef itself). You can do this by draining the contents through a colander set over a saucepan. Return the beef to the casserole. Press any remaining juices from the solids into the cooking liquid, then skim off any visible fat. Boil down the liquid until it reduces to about 3 cups.
Off the heat, whisk in the beurre mani (a mixture of flour and softened butter, which you would have prepared separately). Return the sauce to a simmer for about 2 minutes, stirring constantly, until it thickens lightly and coats the back of a spoon. Taste and adjust the seasoning as needed.
Pour the thickened sauce over the beef in the casserole, then gently fold in the cooked pearl onions and sautéed mushrooms. To serve, bring the stew back to a gentle simmer, basting the meat and vegetables with the rich sauce for several minutes until everything is heated through.