Remove the steak from the fridge at least 1 hr before cooking and let it come to room temperature. The steak should have an even 1cm thickness, so lightly bash any thicker areas with a meat hammer or rolling pin. Scatter the salt over both sides of the steak and set aside.
To prepare the chips, bring a large pan of water to a simmer and cook them for about 10 mins. Test with a fork after 6 mins – they should be very tender with slightly rough edges. If they’re not yet ready, cook for the remaining time and check again. Carefully transfer to a wire rack using a slotted spoon and leave to steam-dry. You want them to be as dry as possible before frying.
Next, make the sauce. Melt the butter in a small frying pan until foaming, then fry the shallots and garlic until soft, around 5 mins. Add the mushrooms and cook for 1-2 mins more until they release some of their moisture. While the veg is cooking, bring the cream and tarragon stems to a simmer in a small pan. Pour through a sieve into the pan with the mushroom mixture. Stir, then reduce the heat and continue to cook the sauce until slightly thickened and smooth. Remove from the heat and set aside.
To fry the chips, tip the oil into a deep pan, making sure the pan is no more than two-thirds full, and ideally with a frying basket. Heat the oil to 180C. If you don’t have a thermometer, you can check the temperature by dropping in one of the chips – if it sizzles immediately, it’s ready. Deep-fry the chips in batches, making sure you don’t overcrowd the pan, for 5-6 mins per batch until golden and crisp at the edges. Tip out of the frying basket onto kitchen paper (or remove with a slotted spoon) and leave to drain. Keep warm in a low oven while you cook the steak.
Heat a heavy-bottomed frying pan over a high heat until smoking hot. Pat the steak dry with kitchen paper (the salt will have drawn out some moisture), then drizzle with the oil. Cook for 3-4 mins on each side for medium-rare, or until cooked to your liking. Transfer to a board or plate and leave to rest – keep warm by covering with foil. Pour any resting juices into the sauce.
Reheat the sauce, then, using a very sharp knife, carve the steak across the grain into 1-2cm slices (or four equal steaks, if you prefer). Drizzle the sauce over the steak, scatter over the tarragon leaves, and serve with the chips and watercress on the side.