How to Make It
Light the grill or heat the broiler. Coat the chicken breasts with 1 tablespoon of the oil and sprinkle with 1/4 teaspoon each of salt and pepper. Grill or broil the chicken for 5 minutes. Turn and cook until just done, about 5 minutes more. When the chicken breasts are cool enough to handle, cut them into bite-size pieces.
Toss the asparagus spears with 1 1/2 tablespoons of the oil and 1/8 teaspoon each of salt and pepper. Grill or broil the asparagus, turning occasionally, until tender, about 10 minutes, depending on the width of the spears. Cut the spears into 2-inch lengths.
In a blender, combine the garlic, water, parsley, lemon juice, 1/2 teaspoon salt, and the remaining 1/3 cup oil. Puree until smooth, scraping down the side of the blender with a spatula as necessary.
In a large glass or stainless-steel bowl, toss the lettuce, chicken, and asparagus with half the vinaigrette. Put the salad on plates. Drizzle the remaining vinaigrette over the salads.