Prepare the Salmon: Pat the salmon fillets dry with paper towels. Season both sides with salt and pepper.
Sear the Salmon: Heat 1 tablespoon of butter and olive oil in a large nonstick skillet over medium-high heat. Once the butter is melted and foaming, place the salmon in the skillet, skin-side up if applicable. Sauté for 3–4 minutes, or until the first side is golden brown. Flip the fillets, reduce the heat to medium, and cook for an additional 3–4 minutes until fully cooked and flaky. Transfer the salmon to a plate and cover loosely with foil.
Cook the Onion and Make the Sauce: In the same skillet over medium-low heat, add the chopped onion and sauté for 3 minutes until softened. Deglaze the pan by stirring in the chicken broth, lemon juice, and capers, using a wooden spoon to scrape up any browned bits. Bring the mixture to a boil over medium heat and simmer for 3–5 minutes until the sauce thickens slightly.
Finish the Sauce: Remove the skillet from the heat and whisk in the remaining 3 tablespoons of butter, adding one tablespoon at a time to create a smooth, emulsified sauce.
Combine and Serve: Taste the sauce and adjust the seasoning with salt and pepper if needed. Return the salmon fillets to the skillet, spooning the sauce over each. Garnish with parsley and lemon slices if desired. Serve immediately with your choice of side dishes.