Preheat the oven to 200C/400F/Gas 6. Season the steaks all over with salt and freshly ground black pepper.
Heat a large ovenproof frying pan over a high heat until very hot. Add a little of the butter and one tablespoon of the oil. When the butter is foaming, add the steaks and fry for 1-2 minutes on each sides, or until browned on both sides.
Transfer the steaks to the oven and roast for 8-10 minutes (for rare), 12-15 minutes (for medium), or until cooked to your liking.
Remove the steaks from the oven and set aside to rest for at least 5 minutes.
Meanwhile, blend 75g/3oz of the butter and all of the blue cheese in a food processor until just smooth. Season, to taste, with salt and freshly ground black pepper.
Add half of the chives and stir through until combined.
Heat an ovenproof frying pan over a medium to high heat until hot. Add the remaining butter and olive oil and fry the blanched potatoes for 2-3 minutes, or until golden-brown and crisp. Add the girolle mushrooms and fry for a further 2-3 minutes.
Season the mixture, to taste, with salt and freshly ground black pepper, then stir through the parsley and the remaining chives.
To serve, place one côte de boeuf onto each of two serving plates. Spoon some of the cheese butter over the top, then pile the mushrooms and potatoes alongside.