1-First, pat the salmon fillets dry and season them generously with salt and pepper to enhance the flavor. Heat the olive oil in a large cast-iron skillet over medium-high heat, then place the salmon fillets skin-side up in the skillet and add 1 tablespoon of butter. Cook for 3 minutes until golden, flip with a spatula, reduce heat to medium, and cook for another 3 minutes.

2-Remove the salmon from the skillet and set it aside, then drain all but 2 tablespoons of the oil and butter mixture. Add the diced shallot and minced garlic, cooking and stirring constantly for 30 to 45 seconds until fragrant. Pour in the dry white wine and add the capers, bring to a simmer, and cook until only 1 to 2 tablespoons of liquid remain.

3-Remove the skillet from heat and swirl in the remaining 7 tablespoons of butter, one piece at a time, until the sauce is fully incorporated and emulsified. Stir in 1 tablespoon of the minced fresh parsley and adjust seasoning with lemon juice, salt, and pepper to taste. Finally, return the salmon to the skillet to warm through in the sauce and garnish with the remaining 1 tablespoon of minced fresh parsley before serving immediately.