The base of this fully loaded Caesar salad is an easy dressing made with pantry staples. Then we amp up the flavor by grilling all the components, adding a delicate smokiness. If you've never used cedar planks for salmon, here is your chance. We're pretty sure you'll never go back to the hassle of fish sticking to the grill grates again.
Soak two 5 1/2-by-11-inch cedar planks in water in a large bowl for 30 minutes before grilling to avoid burning.
Prepare a grill for medium indirect heat (for a gas grill, leave one or two burners on low or off; for a charcoal grill, bank the coals on one side of grill).
Whisk together the mayonnaise, Worcestershire sauce, anchovy paste, mustard, garlic, lemon juice, 1/2 teaspoon salt and several grinds of pepper in a medium bowl. Remove 2 tablespoons to a small bowl; cover and refrigerate the remaining dressing.
Pat the cedar planks dry with a paper towel and rub each side lightly with vegetable oil to coat. Put 2 salmon fillets on each plank and brush evenly with the 2 tablespoons dressing. Sprinkle with the Parmesan and a generous amount of salt and pepper.
Put the cedar planks over indirect heat, cover the grill and grill until the salmon is cooked through and the cheese is melted, about 8 minutes depending on the thickness of your salmon. Transfer the cedar planks with the salmon to a baking sheet using tongs. Set aside.
Put the bread, lettuce, tomatoes stem-side up and avocados cut-side down over direct heat. Grill, flipping the bread and lettuce once, until lightly charred, 5 to 7 minutes; transfer the ingredients to a large platter as done.
Transfer the fish to the platter with a spatula. Drizzle everything with the reserved dressing and sprinkle with the parsley. Shave some Parmesan over the top with a vegetable peeler.