Pat the salmon fillets dry using paper towels to remove any excess moisture. Season both sides evenly with kosher salt and a few grinds of black pepper.

In a 10-inch nonstick or cast-iron skillet, melt 2 tablespoons of butter over medium heat. When the butter begins to foam, add the salmon fillets, skin-side up. Sear for approximately 3 minutes until the surface turns golden brown. Gently flip and cook for an additional 3 minutes, or until the skin is lightly crisp. Transfer the salmon to a plate and set aside. (The salmon will finish cooking in the sauce.)

Add the minced garlic to the same skillet and sauté briefly until fragrant, about 30 seconds. Pour in the dry white wine, scraping any browned bits from the bottom of the pan with a wooden spoon. Simmer for 1 minute, allowing the liquid to reduce slightly.

Reduce the heat to medium-low. Add the lemon zest, lemon juice, Dijon mustard, and capers. Stir well to combine. Add the remaining 2 tablespoons of butter and continue stirring until melted and the sauce is smooth. Taste and adjust seasoning with salt and pepper as needed.

Place the seared salmon back into the skillet, skin-side down. Spoon the sauce over the fillets and simmer for 1–2 minutes, or until the salmon is just cooked through. The internal temperature should reach 120°F to 130°F for medium-rare doneness.

Transfer the salmon to serving plates. Spoon the lemon-butter caper sauce over the top and garnish with freshly chopped parsley if desired. Serve immediately with your preferred side dish.