To make the Bearnaise sauce, mix together the vinegar, water, peppercorns, bay leaf, shallot and sprig of tarragon in a small saucepan. Reduce the mixture to 1 tbsp, then strain into a bowl.
Set a heatproof bowl over a pot of barely simmering water and add the egg yolks. Make sure the the bowl doesn’t touch the water.
Whisk together the yolks and add a little salt, 1/2 tsp of the reduction liquid and 1 cube of butter. Continue whisking until the sauce slightly thickens.
Add the remaining butter to the sauce one cube at a time and continue whisking until thoroughly combined. The sauce will lose some volume but should still have a thick, aerated texture.
If the sauce begins to separate or becomes too oily, add a few drops of cold water and it will come back together.
Add the remaining reduction and chopped tarragon and remove the bowl from the pan. Keep the sauce warm in the bowl until ready to serve. If the sauce is too thick, add 1tsp of cold water to thin it out.
Rub the steaks with the oil and season to taste. Heat a chargrill pan over a high heat and cook the steaks for 2-3 minutes each side for medium-rare.
Cover the steaks with foil and leave to rest for 5 minutes.
Serve the steaks with the Bearnaise sauce on the side and thickly cut fries, if liked.