Pat the 2 large salmon fillets dry with paper towels. Season both sides with 1 teaspoon of the salt and black pepper.
Heat the 2 tablespoons olive oil in a large skillet over medium-high heat. Once the oil is shimmering, add the salmon skin-side up (or presentation side down if skinless). Cook undisturbed for about 4 minutes, until a golden crust forms.
Carefully flip the salmon, reduce the heat to medium, and cook for another 1 to 2 minutes, or until the salmon reaches your preferred doneness. Transfer the fillets to a plate and set aside.
In the same skillet, melt the 4 tablespoons salted butter. Add the 1 finely chopped shallot, 1 clove minced garlic, and remaining ½ teaspoon salt. Cook for about 1 minute, stirring frequently, until the shallot softens and the garlic is fragrant.
Stir in the ½ cup chicken broth or vegetable broth, 4 tablespoons lemon juice, 1 teaspoon lemon zest, and 3 tablespoons capers. Bring the sauce to a gentle simmer and cook for about 4 minutes, allowing it to reduce slightly.
Return the salmon to the skillet and spoon the sauce over the top. Let it warm through for about 1 minute.
Sprinkle with 2 tablespoons finely chopped Italian parsley and serve immediately with lemon slices, if desired.