Yes! But sometimes, no! Basically it depends.
Many dictionaries will give parmesan as the translation of Parmigiano. There is good reason for this: in 2008, the European Court of Justice established that only the certified product known as Parmigiano Reggiano DOP could be sold under the English language name “Parmesan” in the EU.
And yes, Parmigiano is supposed to be capitalized, as it is a proper noun. Most other cheeses -even gorgonzola or pecorino romano – do not receive the same treatment from Italian grammar.
However, in the USA, Parmigiano Reggiano is sold in stores under its Italian name, alongside canisters of grated parmesan cheese or parmesan wedges that were produced in North America.
Parmesan cheeses produced outside of Italy are not subject to the EU’s rules. They are more or less imitations of Parmigiano Reggiano, but the difference is in the details: saltiness, texture, length of the aging process, raw ingredients used.
To sum it up: if you’re buying groceries in Europe, then the parmesan you see on the shelf is likely to be Parmigiano Reggiano DOP.
If you’re shopping in an American supermarket, however, parmesan and Parmigiano are two different cheeses entirely.
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