Make the Caesar dressing. Combine ¾ cup mayonnaise, 2 Tbsp basil pesto, 2 anchovies, 2 Tbsp parsley, 2 Tbsp lemon juice, 1 Tbsp grated Parmesan, 1 clove of garlic, and 1½ tsp Dijon mustard in the bowl of a food processor fitted with blade attachment. To make the dressing more pourable, I like to add 1-2 Tbsp water, but this is optional. Process until smooth and creamy, about 30 seconds, scraping down the sides as needed.
Make the croutons. Preheat an oven to 375°F. Place ripped focaccia pieces on a rimmed baking sheet, then drizzle with 1-2 Tbsp olive oil and a generous sprinkle of kosher salt. Bake for 5-7 minutes, or until crispy and golden brown. Set aside.
Season the salmon. Pat salmon fillets dry on both sides, then season liberally with 1 tsp kosher salt and a little black pepper.
Cook the salmon. Heat a large skillet over medium-high heat until hot, about 2-3 minutes. Coat with 2 Tbsp of neutral oil. Carefully place each fillet (skin-side down, if using skin-on) into the hot pan. The sizzle you hear should be moderate–if it’s too loud or popping, reduce the heat slightly. Allow the salmon to cook undisturbed for 4-5 minutes, until it easily lifts away from the pan. Once the edges turn opaque about halfway up the sides, use a spatula or tongs to carefully flip. Cook on the second side for 1-2 minutes more, or until an internal temperature of 130-135°F (54-57°C) is reached. Transfer to a plate.
Assemble, then serve: In a large salad bowl, combine chopped kale, romaine lettuce, ½ cup of shredded Parmigiano Reggiano, and the focaccia croutons. Spoon in as much of the pesto Caesar dressing as you like, then toss thoroughly. Transfer the salad to individual plates, then top each with a mountain of shredded Parmigiano if you’d like more. Place one salmon fillet on each portion, sprinkle with a pinch of flaky sea salt, then enjoy!