Makes 4 servings.
Cut tips off 12 asparagus spears and reserve. Cut all spears into 1/2-inch pieces and cook in a saucepan of salted boiling water until tender, about 6 minutes. Transfer cooked asparagus with a slotted spoon to a blender and purée with 1 cup cooking liquid, reserving remaining cooking liquid in pan. Keep purée warm in a small saucepan.
Have ready a bowl of ice and cold water. Return reserved cooking liquid to a boil and blanch reserved asparagus tips until just tender, about 3 minutes. Drain tips and immediately transfer to ice water to stop cooking. Drain tips and reserve for garnish.
Scrub potato and cut lengthwise into four 1/2-inch-thick slices. In saucepan cook potato slices in salted boiling water to cover until tender, about 5 minutes, and transfer with a slotted spoon to a bowl. Keep potatoes warm, covered.
Pat snapper dry and season with salt and pepper. In a large non-stick skillet heat oil over moderate heat until hot but not smoking and cook fish, skin sides down, about 3 minutes, or until skin is golden brown. Turn fish over and cook 2 minutes more, or until just cooked through.
Whisk lemon juice and salt and pepper to taste into asparagus purée.Spoon some sauce onto 4 plates and top each portion with a potato slice and a fish fillet. Garnish fish with asparagus tips and serve remaining sauce on side.