Preheat the oven to 170°C/325°F/Gas Mark 3. 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 clingfilm and flatten out with a steak hammer or rolling pin to a 3 mm thickness. Place a piece of Parma ham on top of each slice of beef, trimming them down as necessary and then spread the stuffing on top. Roll each one in a Swiss roll shape and secure with toothpicks.
Heat a heavy-based casserole dish 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 casserole dish and pour in the wine. Add 50 ml of water, then cover and cook in the oven for 1½ 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 200°C/400°F/Gas Mark 6 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. Roast for 8-10 minutes or until bubbling and lightly golden.
Arrange the stuffed braised beef on warmed plates, spoon over a little of the remaining juices and serve with a generous helping of the crushed new potato and Parmesan gratin.