American 'cup' measurement — for cheese

Technically, anything after a comma in an ingredients list should happen after measuring. As this answer on the cooking stackexchange says,

"1 cup of chopped nuts" is measured after chopping.

"1 cup of nuts, chopped" is measured before chopping.

Any proper cookbook or professional recipe will follow this convention. However, in this day and age, anybody can post a recipe to a website, and many such people seem to be unaware of the convention. In your example, for instance, you can't really measure cheese by the cup until after it's shredded¹.

To descend into cooking advice rather than language advice: shred the cheese, then measure it. What's the worst that can happen? You add too much cheese? You do know that there's no such thing as "too much cheese", don't you? :)

^{1 Note that shredded ≠ grated. Any solid cheese can be shredded. Only a very hard cheese can be successfully grated.}

Other cooks have had your same puzzle. The processed food industry has helpfully placed both measures of weight and volume on many of its packages.

Take a look at this Kraft cheese package.

This is one pound (weight) of cheddar cheese, and the packaging indicates that it yields about 4 cups (volume) of shredded cheese.

To get "one cup of cheese, shredded," you may also weigh a 4 oz. block of cheese and shred it. You probably have a four-sided grater such as this one.

To shred cheese, use the bigger holes (on the right face of the picture). To grate cheese, use the smaller holes (on the left face of the picture). As others have mentioned, only harder cheeses (like Parmesan or Gouda) may be grated.

Or, to follow the "Joy of Cooking" (sorry, citing from memory here), shred a block of cheese and lightly tamp in down, filling an 8 ounce dry measure.

A "cup" is in fact a measure of volume, so when you are looking to make a recipe that calls for 1 cup of shredded cheese, you want to take a dry measuring cup and fill it with shredded cheese. Which seems like a lot to me, but I suppose some recipes could call for it. Recipes that call for shredded cheese are rarely that precise.

Also, try this question: What does "cup" mean in "cup of cheese"?