Season salmon fillets with kosher salt and pepper.
Heat avocado oil in a large non-stick skillet over medium-high heat until hot and shimmering. Add salmon fillets in a single layer and cook undisturbed until golden and crisp on one side, 3-4 minutes. Reduce heat to medium and, with a thin spatula, flip each fillet and cook 1-2 minutes more until nearly done (they'll continue to cook on the plate and be returned to the pan for one minute, so take care not to overcook). Transfer salmon fillets to clean plate.
Add broth, lemon juice, and capers to skillet and bring to boil, scraping up any browned bits; boil until reduced slightly, 1-2 minutes. Add butter and stir to melt. Return salmon fillets to skillet and spoon sauce over until coated and heated through, about 1 minute (or until salmon is cooked to your liking). Remove from heat and transfer salmon and sauce to platter. Garnish with parsley and lemon slices if desired and serve.
If you prefer to leave the skin on your salmon fillets, sear them skin side down until the skin is crispy, 6 to 7 minutes then flip and cook flesh side down for just 1 to 2 minutes more before transferring to plate.