Salmon Caesar Salad

Name: Salmon Caesar Salad
Ingredients: 1/2 cup mayonnaise; 1 tablespoon Worcestershire sauce; 2 teaspoons anchovy paste; 2 teaspoons Dijon mustard; 1 clove garlic, grated; Juice of 1 lemon; Kosher salt and freshly ground black pepper; Vegetable oil, for the cedar planks; Four 6-ounce center-cut skin-on salmon fillets; 1/2 cup freshly grated Parmesan plus a piece for shaving; 4 thick slices country bread; 2 hearts romaine lettuce, halved lengthwise; One 12-ounce container on-the-vine cherry tomatoes; 2 avocados, halved and pitted; 1/2 cup fresh parsley leaves, chopped