Mash anchovy fillets. Whisk in mashed garlic and mustard. Stir in vinegar and lemon juice; whisk in sugar, salt and pepper. Slowly whisk in oil until thickened. Add half the dressing (about 1/2 cup) to chicken in a plastic bag; seal. Reserve remaining dressing. Marinate chicken at room temperature, 20 minutes.
Heat grill, broiler or grill pan to medium-high heat. Grill chicken until instant-read thermometer registers 170°F, about 3 to 5 minutes per side. Let chicken rest 5 minutes.
In large bowl, toss remaining dressing, Parmesan and croutons. Slice chicken diagonally into thin strips; layer over salad.
