Using largest holes on a box grater, coarsely shred enough cheese to measure 2 1/2 cups.
Toss together cheese and cayenne in a bowl, then make 4 mounds of cheese (1 tablespoon each) about 2 inches apart in a 12-inch nonstick skillet. Cook over low heat (cheese will melt and spread), undisturbed, until bubbling and firm enough to lift, 4 to 5 minutes (do not turn over). Quickly transfer with a metal spatula to a rolling pin, allowing chips to drape over pin, and cool completely.
Pour off fat from skillet and wipe clean with paper towels. Make more chips in same manner.
·You can make flat chips instead of curved ones by transferring them to a rack to cool instead of to a rolling pin.·Chips can be made 3 days ahead and kept, layered between sheets of wax paper, in an airtight container at room temperature.