Combine the milk and the water, bring them to a boil. Add the polenta slowly and continuously wisk so it does not form clumps. Turn the heat to a simmer and continue to wisk until cooked through, about 10 minutes.
When it's almost done (before all of the liquid evaporates), melt the butter in a pot, without letting it crackle, and combine it with the cheese, stirring to obtain an even mixture. Pour half the polenta onto a polenta board or a large serving platter, spread the cheese mixture over it, and cover with the remaining polenta. Serve steaming hot.