To make the béarnaise sauce, place the vinegar, white wine, peppercorns and shallot into a small saucepan and bring to the boil.
Simmer until the liquid has reduced by half.
Remove the peppercorns then pour into a medium bowl.
Add the egg yolks and whisk well, then add the lemon juice.
Place over a pan of simmering water and whisk together until thickened and light in colour.
Gradually add in the melted butter, whisking constantly.
Season with salt and black pepper and add the chopped tarragon leaves.
Turn off the heat and leave the bowl over the pan until ready to use
For the steak, preheat the oven to 200C/400F/Gas 6.
Season the chateaubriand with salt and freshly ground black pepper.
Heat a large ovenproof frying pan until hot, add the oil and half the butter then add the beef and seal on each side until browned, adding the rest of the butter as you go, basting the beef with it.
Transfer to the oven and roast for 15-20 minutes, (depending how you like your beef).
Remove from the oven, turn over and allow to rest for at least 15 minutes.
Meanwhile, place the cherry tomatoes onto a roasting tray and place in the oven for 8-10 minutes until just bursting.
Heat a deep fat fryer to 160C/320F. (CAUTION: Hot oil can be dangerous. Do not leave unattended.)
Carefully place the potato batons into the fat fryer in batches and fry for 3-4 minutes, or until tender but not coloured.
Remove the fries then increase the heat on the fat fryer to 190C/375F and return the fries to the fryer and cook for 2-3 minutes until golden-brown and crisp.
Remove with a slotted spoon and drain onto kitchen paper then season with salt and a little black pepper.
To serve, carve the rested beef into thick slices and place onto the serving plate. Pile the fries and tomatoes alongside then finish with a dollop of béarnaise and handful of watercress.