In short, both “patty” and “paddy” are correct words, but they have different meanings and applications. “Patty” refers to a small, flat, and usually round portion of food, typically made of ground meat or vegetables. On the other hand, “paddy” is primarily used to describe a flooded field in which rice is grown.

Because St. Patrick’s Day is originally an Irish holiday—and Gaelic is a traditional Irish language—the right nickname is Paddy, rather than Patty.

Paddy is correct, because it is based on the Irish boy's name Pàdraig (the English version is Patrick). I learned that the Irish dislike when people write Patty (this is short for Patricia, a girl's name), so be sure to spell it Paddy and St. Paddy's Day from now on.