Pat the fish fillets completely dry with paper towels before seasoning them. This step matters more than people think, since any extra moisture will prevent the fish from browning properly in the pan. Season both sides with salt, pepper, and paprika so the flavor reaches all the way through.
Heat the olive oil in a large skillet over medium high heat until it shimmers. Place the fillets in the pan without crowding them, cooking two at a time if needed. Sear each side for about 3 to 4 minutes until golden brown and just cooked through, then remove and set aside.
Lower the heat and add the butter to the same skillet, letting it melt and turn slightly golden. Stir in the minced garlic and cook for about 30 seconds until fragrant, being careful not to let it burn. Add the lemon juice, lemon zest, and broth, then let the mixture simmer gently.
Let the sauce simmer for 2 to 3 minutes, scraping up any browned bits from the bottom of the pan. This step pulls in all the flavor left behind from searing the fish, making the sauce richer. The sauce should slightly reduce and thicken during this time.
Place the seared fillets back into the skillet, spooning the warm lemon butter sauce over each piece. Let everything simmer together for another minute or two so the fish soaks up the flavor. This keeps the fillets moist while marrying the fish and sauce together.
Sprinkle the chopped parsley over the fillets right before serving for a fresh, colorful finish. Add a few lemon slices on top or around the plate for extra visual appeal. Serve immediately while the fish is warm and the sauce is glossy.