For the steak, season the steaks well with salt and freshly ground black pepper.
Heat a frying pan until smoking, add the butter and olive oil and fry the steaks on both sides for 2-3 minutes (for medium), or until cooked to your liking.
Heat the vegetable oil a deep-based pan to 190C/375F, checking with a food thermometer. (CAUTION: hot oil can be dangerous. Do not leave unattended.) Add the matchstick potatoes and fry for 3-4 minutes, or until golden-brown and cooked through. Remove with a slotted spoon and drain on kitchen paper, then season, to taste, with salt and freshly ground black pepper.
For the sauce, heat a clean frying pan until smoking, add 25g/1oz of the butter and fry the shallots for 2-3 minutes, or until golden-brown. Add the mushrooms and garlic and cook for one minute, then pour in the wine and simmer for 2-3 minutes.
Add the chicken stock and simmer for 10-12 minutes, or until the liquid has reduced by half. Whisk in the remaining butter until the sauce is smooth and creamy. Stir in the tarragon leaves and season, to taste, with salt and freshly ground black pepper.
To serve, pile the fries onto serving plates, place the steak alongside and spoon over the sauce.