Heat a large nonstick skillet over medium-high heat. Dry the salmon fillets well with paper towel and season evenly with 1 teaspoon salt.
Add the oil to the pan and heat an additional 30 seconds. Add the fillets to the pan, flesh-side down, and cook undisturbed until deep golden brown, about 4 minutes. The fish should look cooked up the sides of the fillets. Carefully flip the fish and cook an additional minute or until a thin metal skewer inserted in the thickest part comes out warm when touched to your bottom lip. Using a slotted spatula, remove the fish to a plate to rest.
Return the pan to medium heat and add the shallot and remaining 1/2 teaspoon salt, adding more oil if needed. Cook, stirring often, until fragrant and soft, about 1 minute. Add the chicken broth, lemon juice, capers and lemon zest. Simmer for 2 minutes to reduce slightly. Whisk in the butter and mustard until smooth and combined. Stir in the parsley and spoon the sauce over the salmon.