4 (5- to 6-ounce) salmon filets, preferably wild
Kosher salt and freshly ground black pepper
1/3 cup plus 3 tablespoons extra-virgin olive oil, divided
4 cups baby arugula
2 tablespoons fresh juice from 1 lemon
Grated Parmigiano, for serving
1 ripe Hass avocado, pitted and cut into 1/2-inch cubes
Press salmon filets between paper towels to dry surfaces thoroughly. Season on all sides with salt and pepper. Heat 1/3 cup oil in a large stainless steel skillet over medium-high heat until shimmering. Add salmon filets skin side-down. Immediately reduce heat to medium-low. Cook, pressing gently on back of filets to ensure good contact with skin, until skin is rendered and crisp, about 6 minutes. If skin shows resistance when attempting to lift with a spatula, allow it to continue to cook until it lifts easily.
Meanwhile, in a large bowl, combine arugula, lemon juice, remaining 3 tablespoons olive oil, and avocado. Season with salt and pepper and toss in grated cheese.
Flip salmon and cook on second side until a thermometer inserted into the thickest part registers 120°F for medium rare or 130°F for medium, about 1 minute longer. Transfer salmon to a paper towel-lined plate. Spoon salad onto plates, top with salmon, and serve