Preheat oven to 350 degrees. Test each of the three eyes at stem end of coconut to see which is the softest. Then use an ice pick (or a screwdriver) and a hammer to pierce two of the eyes. Strain milk through a fine sieve into a bowl; reserve for another use or discard.
Place coconut on a rimmed baking sheet; bake for 30 minutes, or until coconut shell begins to crack. Set aside until cool enough to handle.
Wrap coconut in a clean kitchen towel; holding coconut with one hand, hit it with hammer in the same place several times to crack the outer shell and split coconut into several large pieces.
Separate coconut flesh from shell, and use a vegetable peeler to remove dark outer skin. Rinse coconut in cold water, then use vegetable peeler to take off long, thick strips from the outer (cut) edges. Use immediately, or store, layered between damp paper towels in an airtight container, in the refrigerator for up to 6 hours.