Asparagus comes in all different thicknesses, from pencil-thin to nearly an inch. The bottom woody inch or two of any asparagus should be discarded. Whether or not you peel the stalks depends upon their thickness. The larger the asparagus, the tougher the peel, and the more likely they are to need peeling. I usually peel the stalks if the asparagus is 1/2-inch in diameter or larger. Large asparagus stalks tend to be much thicker at the bottom than at the top, so peeling makes them more uniform and they will cook more evenly. To keep the stalk from snapping during peeling, place it on a flat surface and run a vegetable peeler from just below the tip all the way to the bottom.