Make the sauce: Set a large skillet over medium heat and put in 2 Tbs. of the butter. Add the shallots and sauté until tender, 1 to 2 minutes. Add the mushrooms and cook until lightly browned, about 7 minutes. Add the wine, cider, and cider vinegar; raise the heat to medium high and reduce the liquids by about three-quarters. Add the beef broth and again reduce by half. Take the pan off the heat. Add the rosemary, whisk in the remaining 2 Tbs. butter (if you make the sauce ahead, wait to add the butter until just before serving), and taste for seasoning.
Prepare the beef: Season the filet with pepper. Fold the thin tail piece over itself, and if needed, shape the larger head piece into as long a shape as possible. Wrap the bacon strips around the filet, leaving some space between each strip (don’t be tempted to cover the filet completely with bacon; too much bacon will overwhelm the flavor of the beef). Using butcher’s twine, tie the filet at regular intervals to keep the bacon in place and to even the shape of the beef as much as possible. If you’re working ahead, refrigerate the filet until half an hour before you’re ready to roast it. Heat the oven to 375°F. Season the meat liberally with salt and pepper. Tuck the rosemary sprigs under the twine along the length of the filet. Set the roast on a rack in-a roasting pan or rimmed baking sheet and put it in the oven. If there are drippings, periodically use a pastry brush or a spoon to remove excess fat that renders out of the bacon and brush it onto the roast (the rendered fat may cause your oven to smoke a bit). If the bacon begins to brown too quickly, loosely tent the filet with foil for the rest of the cooking time. After 40 minutes, insert an instant-read thermometer deep into the thickest part of the meat; you’re looking for a temperature of 128°F for medium rare (remember that the filet will continue to cook as it rests). Depending on its thickness, the filet should take 45 minutes to 1 hour. Remove the filet from the oven, tent it with foil, and let it rest for 10 minutes before slicing.
Tie the filet at ten intervals bacon slices and to create a neat package.
Tuck sprigs of rosemary under the twine to add extra flavor during roasting.
To serve: Reheat the sauce gently and thin it with a little water, if needed. Whisk in the rest of the butter. Cut the beef into 3/4-inch slices and arrange on plates with the celery root purée. Drizzle on the sauce and serve.