The traditional Asturian dish is La Fabada. It's made with Faves, which are large white beans, that are cooked with chorizo, morcilla (blood sausages) and lacon (thick bacon). It's delicious and stocky and will probably heal whatever ails you.
If you go to Asturias, eat the Fababa. You won't regret it!
In Cangas de Onis the most popular alcoholic drink is cider (La sidra). You can try this drink in special bars "sidreria". Don't miss the possibility to look how to fill cider in glasses. I read somewhere that not every barman can do it, there are special schools for cider-barmen. It is an art!