Preheat the oven to 350°F. Soak the bread in water for 1 minute, then squeeze dry and finely chop. Tip into a bowl and add the onion, carrot, celery, parsley, Parmesan, egg and thyme. Season to taste and mix well to combine.
Cover the steak slices with plastic wrap and flatten out with a steak mallet or rolling pin to 1/8 inch thickness. Place a piece of prosciutto on each slice of beef, trimming them down as necessary and then spread the stuffing on top. Roll each one in jelly roll shape and secure with toothpicks.
Heat a heavy-based Dutch oven or pot that has a lid. Add the oil and quickly sear the beef rolls on all sides until lightly golden – you may have to do this in batches depending on the size of your dish.
Arrange all the beef rolls in the Dutch oven and pour in the wine. Add ¼ cup of water, then cover and cook in the oven for 1 1/2 hours or until the beef rolls are completely tender, adding a little more water if at any time it looks like the beef is going to cook dry.
Meanwhile, preheat a separate oven to 400°F to make the gratin. Cook the potatoes in a pan of boiling salted water for 15-20 minutes until tender. Drain well and return to the pan, then roughly crush with a fork.
Tip into a baking dish. Mix together the Parmesan and parsley and sprinkle over the potatoes, then drizzle the olive oil on top. Bake for 8-10 minutes or until bubbling and lightly golden.
Arrange the stuffed braised beef on warmed plates and spoon over a little of the remaining juices. Place the crushed potato and Parmesan gratin directly on the table and allow guests to help themselves.