Step 1Heat a grill or stovetop grill pan sprayed with nonstick cooking spray.
Step 2Combine the pressed garlic, mayonnaise, anchovy paste and lemon juice. Set aside.
Step 3Lightly season the chicken breasts with salt. Cut 1 garlic clove in half; rub the chicken breasts with both halves. Spray the chicken with nonstick cooking spray, then grill it about 10 minutes. Turn the chicken and grill until the chicken breasts are browned and no longer pink in the center, about 10 minutes. Place the chicken on a platter and keep it warm.
Step 4Cut the remaining garlic clove in half and use it to rub the bread slices. Lightly spray one side of the slices with nonstick cooking spray and place the bread sprayed-side down on the grill until lightly browned, about 2 minutes. Turn, spray the other sides with cooking spray, and grill until lightly browned.
Step 5Toss together the romaine and Parmesan, then set aside.
Step 6Spread the grilled bread with the mayonnaise mixture. Slice the chicken breasts into 1/2-inch thick slices and arrange one sliced chicken breast on each of 4 slices of bread.
Step 7Divide the romaine among the sandwiches, placing it on top of the chicken. Top each piece of chicken with an onion round, separating it into pieces, then top with the second slice bread. Cut the sandwiches in half and serve.