Use paper towels to pat the salmon dry. Season both sides of each fillet with the salt and pepper.
In a large nonstick, nonreactive skillet over medium-high heat, combine 1 tbsp of the butter and the oil, swirling the pan as the butter melts to prevent splatter. When the butter has melted and is done foaming, add the salmon skin side up (if your fillets are skin-on) and sauté for 3 to 4 minutes, until the first side is seared and golden brown. Flip the fish, reduce the heat to medium, and sauté for another 3 to 4 minutes, until browned, flaky and done to your liking. Transfer the fish to a plate and tent loosely with foil to keep the salmon warm as you make the pan sauce.
In the same skillet over medium-low heat, add the onion and cook for 3 minutes, until just softened. Stir in the broth, lemon juice and capers using a wooden spoon to scrape up any browned bits from the bottom of the skillet. Increase the heat to medium and bring the mixture to a boil. Cook for 3 to 5 minutes, until the sauce has thickened into a glaze.
Remove the skillet from the heat and stir in the remaining 3 tbsp of butter, 1 tbsp at a time. Stir constantly until the butter is emulsified and incorporated.
Season the sauce with salt and pepper. Return the salmon to the skillet and spoon the sauce over the top. Garnish with the parsley and lemon slices (if using). Spoon the sauce over the fish to serve.