Set a large heavy-based saucepan with 1 tbsp olive oil over a low heat. Add the mushrooms, shallots and thyme and cook gently for 30 minutes, until softened.
Add the pomegranate juice and red grape vinegar, then cook for about an hour, until reduced and sticky.
Add the chicken stock, then simmer for a further hour, until well reduced.
Meanwhile, remove the venison from the fridge and pat dry with kitchen paper. Rub all over with the remaining olive oil, season and leave to come up to room temperature.
Strain the sauce through a fine sieve or muslin cloth, then return it to the pan and whisk in the butter and chocolate until you have a rich glossy sauce. Keep warm over a low heat.
Set a frying pan over a high heat and add the venison. Sear on all sides until cooked to your liking: about 6 minutes for rare, 8 minutes for medium and 9-10 minutes for well done, depending on the thickness of the meat.
Leave the venison to rest for 5 minutes. Cut into thick slices and serve with the chocolate sauce, buttered greens and olive oil mash.