Put the ice in a bowl of cold water. Shave the asparagus into long strips using a vegetable peeler (you may need to thinly slice the thick ends with a knife) and put into the iced water. Put ½ the basil leaves into a mortar with ½ the pine nuts and the garlic. Season, then pound with a pestle until slightly broken down. Finely grate in half the cheese, then muddle together with the olive oil and lemon juice (you are looking to make a crude and loose pesto); season if needed.
Drain and dry the shaved asparagus, then layer up with the rocket and pesto dressing in a large salad bowl; toss lightly but thoroughly to combine. Toss through the remaining basil leaves and pine nuts, then shave over the remaining cheese. Serve straight away.
