For the béarnaise sauce, place the vinegar, white wine, peppercorns and shallot into a small saucepan and bring to the boil.
Simmer until the volume of the liquid has reduced by half.
Strain the sauce through a fine sieve into a medium sized bowl and allow to cool slightly.
Add the egg yolks to the vinegar reduction and whisk well.
Place the bowl over a pan over simmering water (do not allow the bottom of the bowl to touch the water) and whisk constantly 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. Add lemon juice, to taste.
Turn off the heat and leave the bowl over the pan until ready to use.
For the fries, heat a deep fat fryer to 160C/325F. Alternatively, heat vegetable oil in a deep-sided, heavy bottomed pan until a breadcrumb gently cooks and gradually turns golden-brown. (CAUTION:hot oil is 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 from the fryer in their basket, then increase the heat on the fat fryer to 190C/375F (If using a pan, heat the oil until a breadcrumb sizzles and turns golden-brown when dropped in) and return the chips to the fryer and cook for 2-3 minutes until golden-brown and crisp.
Remove the fries from the fryer and drain onto kitchen paper then season with salt and a little black pepper.
For the steaks, heat a griddle pan until hot.
Drizzle the oil over the steaks and rub well to coat. Season with salt and black pepper, then cook on the griddle for 2-3 minutes on each side.
Remove the steaks from the griddle and rest for three minutes before serving.
To serve, place the steaks onto four serving plates, pile the fries alongside and finish with a dollop of béarnaise sauce.