Preheat the oven to 180C/160C Fan/Gas 4.
Place the côte de boeuf into a roasting tray and roast for 30-40 minutes, until cooked to your liking.
While the meat is cooking, prepare the chips.
Heat the oil in a deep-fat fryer to 140C/285F. Alternatively, heat the oil in a deep, heavy-based frying pan until a breadcrumb sizzles and turns very pale golden-brown when dropped into it. (Caution: hot oil can be dangerous. Do not leave unattended.)
Carefully lower the chips into the hot oil, in batches, and fry for 4-5 minutes, or until pale golden-brown but tender. Remove from the hot oil using a slotted spoon and set aside to drain on kitchen paper.
Remove the pan of oil from the heat but do not discard the oil.
When the côte de boeuf is cooked to your liking, remove it from the oven and set aside to rest for 5-10 minutes.
Heat 50g/1¾oz of the butter in a large frying pan over a high heat. When the butter is foaming, add the beef and fry for 1 minute on each side, or until browned all over, spooning the melted butter over the meat as it fries. Set aside to rest again.
For the béarnaise, melt the remaining butter in a saucepan over a medium heat. Skim off any milk solids that rise to the surface, to clarify the butter.
Bring the vinegar to the boil in a separate, non-reactive saucepan with the peppercorns and shallots. Reduce the heat until the mixture is simmering and simmer over a low heat for 3-4 minutes, or until most of the liquid has evaporated, leaving only about 1 tablespoon in the pan.
Whisk the egg yolks in a bowl, then gradually add the melted butter a little at a time, whisking after each addition – the aim is for the mixture to emulsify and thicken slightly. Discard any milk solids at the bottom of the pan.
When all of the butter has been incorporated into the eggs, spoon in the shallots and any liquid remaining in the pan, then stir in the chopped tarragon and season, to taste, with salt and freshly ground black pepper. Keep warm.
Just before serving, return the pan of oil to the heat and heat to 190C/375F (a bread crumb will sizzle and go dark brown in 1 minute). Lower the cooked chips into the hot oil and cook for 3-4 minutes, or until golden-brown and crisp. Remove from the pan using a slotted spoon, set aside to drain on kitchen paper, and sprinkle with salt.
To serve, carve the côte de boeuf into slices and divide them among the serving plates. Pile the chips alongside, then finish with a dollop of béarnaise and garnish with the watercress leaves.