Pat the salmon fillets dry with paper towels and season both sides with half of the kosher salt (1/2 teaspoon in total).
In a shallow dish, whisk together the all-purpose flour, the remaining 1/2 teaspoon kosher salt, and the ground black pepper.
Dredge each salmon fillet in the flour mixture, shaking off excess.
Heat a large skillet over medium heat and add the unsalted butter. Once the butter foams and begins to lightly brown, add the minced shallot and cook until softened, about 1 minute.
Carefully add the dredged salmon fillets to the skillet and cook without moving until the undersides are golden, about 3 minutes.
Flip the fillets and cook for about 2 more minutes, until cooked through to your liking.
Reduce the heat to medium-low. Add the lemon zest, lemon juice, and drained capers to the pan, spooning the browned butter and shallots over the salmon to form the sauce; heat gently for 30–60 seconds to meld flavors.
Transfer the salmon to plates, spoon the sauce over the fillets, garnish with chopped fresh parsley, and serve immediately.
