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Pat your salmon fillets dry and seasoning with 1 teaspoon salt and pepper to taste.
Heat the oil in a 12-inch skillet or sauté pan over medium-high heat until hot and shimmering. Cook the salmon until golden and crisp on the outside, about 4 minutes. Resist the urge to touch with the fillets as they cook. Carefully flip the fillets and reduce the heat to medium.
Continue cooking until done to your liking, 1 to 2 minutes more, depending on how well done you like your salmon. Using a slotted spatula, remove the fish to a plate to rest.
Return the pan to medium heat and add butter, shallot, garlic, and the remaining 1/2 teaspoon salt. Cook, stirring often, until fragrant and soft, about 1 minute.
Add the chicken broth, lemon juice, capers, and lemon zest. Simmer for 4 minutes to reduce slightly. Place salmon back in the pan and heat for an extra minute. Then, spoon the sauce over the salmon. Garnish with parsley and serve warm. Serve with lemon slices, if desired.