Prepare the Fish: Pat the fish fillet very dry with paper towels to remove all moisture. This is crucial for achieving crispy skin.
Season the Fish: Generously salt both sides of the fish, with a little more salt on the skin side to enhance crispiness.
Heat the Pan: Heat a non-stick pan over medium to medium-high heat until about 300°F (150°C). Use clarified butter to prevent burning.
Cook the Skin Side: Place the fish skin side down in the pan, pressing gently to prevent the skin from curling. Cook undisturbed to develop a crispy skin.
Flip and Continue Cooking: Flip the fish carefully once the skin is crispy. Lower the heat slightly and begin basting the fish with hot clarified butter.
Prepare Lemon Segments: While fish cooks, remove the white pith from the lemon and cut out the segments by slicing along the membranes. Remove seeds.
Finish Cooking and Make Sauce: When fish is nearly cooked (internal temperature around 115°F/46°C), replace clarified butter with regular unsalted butter in the pan. Add capers, lemon juice, lemon zest, cracked pepper, and parsley. Baste the fish with this lemon caper butter sauce.
Serve: Remove fish from pan before it overcooks (it will continue cooking off heat). Plate the fish, spoon the sauce around it, garnish with lemon segments and parsley, and serve immediately.