Mince 2 garlic cloves. Juice 1 medium lemon until you have 2 tablespoons. Cut 4 tablespoons unsalted butter into 4 pieces. Pat 4 (6-ounce) salmon fillets dry with paper towels. Season with ½ teaspoon of the kosher salt and ¼ teaspoon black pepper.
Heat 1 tablespoon olive oil in a large (12-inch) nonstick or cast iron skillet over medium heat until shimmering. Place the salmon skin-side up in the skillet, then press down on them with a flat spatula so that there’s a lot of contact with the pan and it browns evenly. Cook without moving, gently pressing down on fish every so often, until the bottom is golden-brown, about 5 minutes. Transfer the salmon skin-side down to a plate (it will not be cooked through).
Add the butter to the skillet. When melted, add the garlic and cook until fragrant, about 30 seconds. Add the lemon juice, remaining ¼ teaspoon kosher salt, and ¼ cup low-sodium chicken broth, and stir to combine.
Return the salmon to the skillet skin-side down. Cook, spooning some of the sauce over the salmon as it cooks, until the sauce is slightly thickened and the salmon is just cooked through, 2 to 4 minutes more. An instant-read thermometer inserted into the middle of the thickest fillet should register 120 to 130°F for medium-rare, or 135 to 145°F if you prefer it more well-done. Taste the sauce and season with more kosher salt as needed. Garnish with chopped fresh parsley leaves if desired.
