Prepare the Oven: Preheat your oven to 400°F.
Mix the Chive Butter: Combine the butter, minced anchovies, chives, garlic, salt, and pepper in a small bowl. Mix until smooth and evenly blended.
Cook the Salmon on the Stovetop: Heat half of the prepared butter mixture in a large oven-safe skillet over high heat. Once melted, place the salmon fillets in the skillet with the skin side facing down. Allow the salmon to cook for about 3 minutes, letting the skin crisp up and occasionally spooning the pan drippings over the fish to enhance flavor.
Add the Capers and Transfer to the Oven: Sprinkle the capers around the skillet (not directly on the salmon) and transfer the entire skillet to the preheated oven. Roast the salmon for about 5 to 8 minutes, or until the fish is just cooked through. Adjust the cooking time based on the thickness of your fillets—thinner cuts will take less time.
Finish the Dish with Butter Sauce: Remove the skillet from the oven. Add the remaining chive butter to the pan, letting it melt into the warm juices. Spoon the buttery mixture over the salmon to coat it thoroughly.
Garnish and Serve: Squeeze the juice from the lemon half over the fish, then garnish with additional minced chives. Plate the salmon individually and serve with the sauce spooned generously over each portion.