Heat the olive oil in a large Dutch oven over medium-high heat. Sear the seasoned beef on all sides to create a crust, 1-2 minutes on each side. Do this in two batches if you must so you don’t crowd the pan and steam the meat. Drain any excess fat from the pot.
Pour a little of the apple cider over the beef and deglaze the bottom of the pan, getting up all the brown bits. Add the rest of the cider, the broth, dried thyme, dried sage, and mustard to the pot with the seared beef. Bring the mixture to a boil, then cover the pot and reduce the heat to low. Simmer, covered, for about 1 ½ hours, stirring occasionally.
Add the chopped potatoes, carrots, mushrooms, celery, onions, fresh rosemary, and fresh thyme to the pot. Increase heat to bring mixture to a boil, then cover pot and reduce heat to low. Simmer the stew until the beef and vegetables are tender, about 30 to 35 minutes.
Whisk together the flour and the water in a small bowl to create a slurry. Stir the slurry into the beef stew and let simmer for 2-3 additional minutes.
Taste Apple Cider Beef Stew and add any additional seasoning (you may need more salt, depending on how sweet your apple cider was).
Serve Apple Cider Beef Stew while hot along with mashed potatoes, crusty bread, polenta, rice, or egg noodles.