Whisk together olive oil, lemon juice, salt and basil. Set aside.
Use a vegetable peeler to shave the asparagus. When it gets to thin to shave, chop. Repeat with remaining asparagus spears.
Place asparagus and avocado in a bowl. Whisk dressing once more, pour over salad. Toss to coat. Season to taste with salt and pepper. Serve immediately.