Pat the salmon fillets very dry with paper towels (this ensures a crisp skin). Season both sides with 3/4 tsp kosher salt and 3/4 tsp black pepper, pressing the seasoning into the flesh.

Place a large heavy skillet over medium-high heat and let it get hot, about 2–3 minutes. Add 1.5 tbsp olive oil and swirl to coat; the oil should shimmer but not smoke.

Carefully place the salmon fillets skin-side down in the skillet (press gently with a spatula for 10 seconds to prevent curling). Cook without moving until the skin is deeply golden and crisp, about 5–6 minutes. Reduce heat to medium if the skin is browning too fast.

Flip the fillets and add 2 tbsp butter to the pan. Spoon the melted butter over the top and cook until the salmon reaches 125–130°F for medium (about 2–4 minutes after flipping, depending on thickness). Remove salmon to a plate and tent loosely with foil to rest 2 minutes.

While salmon rests, melt the remaining 2 tbsp butter in a small saucepan over low heat. Add the minced garlic and cook just until fragrant, about 30–45 seconds (don’t brown). Remove from heat, stir in 1 tbsp lemon juice, lemon zest, and chopped parsley.

If desired, return the skillet to medium heat and add the remaining 1 tbsp lemon juice to deglaze, scraping up brown bits, then pour that and any pan juices into the lemon-herb butter. Spoon the warm butter over the salmon, garnish with lemon wedges, and serve immediately.