In a shallow bowl, combine chicken, half of the cheese, basil, pepper, and salt. Toss well to evenly coat chicken. Turn each piece in cheese mixture to coat completely. Grill chicken or broil on greased rack, 5 inches from heat source in oven until meat in the thickest part is no longer pink, cut to test (15 to 17 minutes). Watch chicken as cheese burns easily. Let cool slightly.
Meanwhile, in a large bowl, whisk oil, lemon juice, anchovy paste, mustard, and garlic to blend. Add lettuce, remaining cheese and croutons; and toss to coat. Divide salad among 8 individual plates.

Cut each chicken breast diagonally into 5 strips; carefully arrange one sliced breast atop each salad. Garnish with additional parmesan cheese and lemon slices, if desired.