Place salmon filets on a paper-towel lined plate to dry. Let sit, then turn over and dry an additional 5-10 minutes. Season salmon on both sides with Kosher salt and black pepper.
Heat oil in a large skillet over medium-high heat. When hot, add salmon skin side down. Cook undisturbed until the skin is crisp and salmon easily pulls away from the pan. Carefully flip and cook the other side until the salmon registers between 110-120F for medium rare.
Transfer the salmon to a large serving plate, then top each with a generous portion of pistachio pesto. Garnish with additional fresh basil leaves and lemon wedges.