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Heat the oven to 160C/140C fan/gas 3. Heat 1 tbsp oil in a heavy ovenproof pan or flameproof casserole dish, big enough to hold the beef cheeks, over a high heat. Dust the beef cheeks in the flour and fry on all sides until brown. Remove from the pan and set aside. Add the carrots, onion and bay, and fry until well coloured. Add the beer and let it simmer and reduce by three-quarters. Return the beef to the pan, cover with the stock and put the pan in the oven, covered. Leave to braise for 3 hrs 30 mins or until tender but not completely falling apart. Once cool, the cheeks can be chilled if making in advance.
Pour the braising liquid through a sieve, return to the pan and put back on a high heat to reduce by half. Return the beef to the cooking liquid and leave to simmer over a low heat before serving.
To make the polenta, heat the milk until steaming. Add the polenta, whisking continuously, and cook over a very low heat for 5 mins. Make sure to stir the polenta at regular intervals so that it doesn’t stick or become lumpy. Add the coffee and 50g butter, season and keep stirring until smooth. Add a little water to loosen, if necessary.
Melt the remaining butter in a frying pan. When foaming, add the mushrooms and garlic, cook for 5 mins, then add the parsley. Cut the beef cheeks in half so that you have four even-sized pieces. Put a few generous spoons of polenta in the bottom of four deep bowls or plates. Put a piece of beef cheek on top and scatter the mushrooms around. Serve the remaining sauce in a jug on the side.