For the béarnaise sauce, place the vinegars, white peppercorns and shallot into a small saucepan and bring to the boil. Simmer until the volume of liquid has reduced by half.
Remove the peppercorns and pour the mixture into a heatproof bowl. Add the egg yolks to the bowl and whisk well.
Place the bowl over a pan of simmering water and whisk the eggs together until pale and thick. Gradually add the melted butter, whisking constantly. Season with salt and freshly ground black pepper and add the lemon juice and chopped tarragon leaves. Turn off the heat and leave the bowl over the warm pan until ready to serve.
For the chips, preheat a deep-fat fryer to 160C/325F. (CAUTION: hot oil can be dangerous. Do not leave unattended.)
Place the chips in the fryer, a few at a time, and cook for 2-3 minutes, or until just tender but not coloured. Drain on a plate lined with kitchen paper and increase the temperature on the fryer to 190C/375F.
Meanwhile, for the rib-eye steaks, season the steaks with salt and freshly ground black pepper. Heat a frying pan until very hot and add the butter and olive oil. When the butter is foaming, add the steaks and fry on each side for 2-3 minutes (for medium) or until cooked to your liking. Remove from the heat, cover and rest in a warm place for at least two minutes before serving.
For the chips, return the chips to the oil and fry for 2-3 minutes, or until golden-brown and cooked all the way through. Remove the chips again using a slotted spoon and set aside to drain on kitchen paper. Season with salt to taste.
To serve, pile the chips on one side of each plate. Place the steak next to the chips along with a spoonful of béarnaise sauce.