Actually, being an italian american person I have to say I don't think there is such a thing as "parmigiana" cheese.
There is parmesean (In italian parmigiano)
Then there are baked italian dishes covered with mozzarella or provolone cheese mixed with parmesean, (and always sprinkled with oregano) that are called "parmigiana." As in... Eggplant Parmigiana, Veal Parmigiana, etc.
But to the best of my knowledge, there is no such animal as a "parmigiana cheese."
So I ask... what the hell are you brits doing screwing up my native language? Jeez, next thing you know you guys will start slapping tomato soup and chedder on a bisquit and call it pizza!
Freaking limey bastards!
