How do you use and store grey mullet bottarga?
Aside from the way it tastes, the best part of bottarga is that it effectively keeps indefinitely. Even after you open up the packaging, peel back the pellicle, and grate some over pasta, the remainder will keep, tightly wrapped in plastic the cut part after usage. Then wrap all of it to an aluminum foil and refrigerate for a year.
A word about that pellicle: It is in fact the membrane that encapsulates the roe sac, which turns papery during the process of curing and drying. While it is not absolutely necessary to remove it, it is preferable.
As Andrew Feinberg notes in Franny's: Simple, Seasonal, Italian, if you don't peel it off, "it might get caught in your teeth." Generally speaking, I peel only as much as I want to use, much like certain salami with inedible chewy casings.
Once it's peeled, you're all set to use it as you see fit. You can use it as a finishing ingredient, finely grated or crumbled over any number of things, or you can enjoy it on its own, in thin slices, dressed with a little olive oil, salt, and a squeeze of lemon.
If you like, you can go in an Asian direction, or a Middle Eastern one; it has been enjoyed the world over for hundreds of years because of its versatility, and it would go well with the flavors of any cuisine.