Preheat the oven to 240°C/475°F/Gas 9. Allow the chateaubriand to come to room temperature then rub all over with the olive oil and sprinkle with sea salt and black pepper.
Heat a heavy-based ovenproof frying pan. Seal the chateaubriand over a high heat for about 6 minutes until golden brown on all sides, turning regularly. Transfer the pan to the oven and roast the steak for about 12 minutes for medium-rare, or to your liking. Remove from the oven and allow to rest in a warm place for 10 minutes.
Meanwhile, prepare the Béarnaise sauce. Melt the butter in a small pan or in the microwave. Combine the white wine, vinegar, one tablespoon of the tarragon, the shallot and white pepper in a pan. Bring to a simmer and reduce until about one tablespoon of liquid remains. Strain into a liquidizer or mini blender, pushing down the tarragon leaves to extract the liquid.
Add one tablespoon of warm water to the liquidizer together with the egg yolks and turn on the machine. Pour the hot butter into a jug and start to pour the butter very slowly on to the egg yolks. As the sauce emulsifies, increase the butter flow to a thin, steady stream.
As the Béarnaise sauce thickens, you will notice a change in the sound of the machine. If the sauce is too thick, add a little more warm water or lemon juice to taste. Add the remaining tarragon leaves, the parsley and cayenne pepper and blend briefly. Season with salt. Keep the sauce warm until you need it in a bowl or jug set in a pan of hot but not boiling water.
Carve the chateaubriand into 12-16 slices, divide between warmed plates, garnish with watercress and serve with Béarnaise sauce on the side and a big bowl of chips.