To Fionn is to eat large quantities of fatty foods or unhealthy food from a fast food outlet (such as McDonalds). The general result is an obese person or a fat ass (most commonly used in reference to males). If you were Fionning, you would be eating the food.
It can be used in context as a noun and as a verb.
Hes eating like a Fionn in a McDonalds!
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