Pat the branzino fillets dry with paper towels, skin side up, removing all moisture.

Season both sides of each fillet with sea salt and black pepper, then let rest for 5 minutes.

Blot any moisture that appears on the surface with fresh paper towels before cooking.

Heat a 12-inch skillet over medium-high heat for 3 minutes until very hot, then add 2 tablespoons olive oil and heat until shimmer.

Carefully place the fillets skin side down in the hot oil and cook undisturbed for 5 to 6 minutes until the skin is deep golden and crisp.

Flip the fillets using a fish spatula, add the remaining 1 tablespoon olive oil around the edges, and add sliced garlic and thyme sprigs.

Cook the second side for 3 to 4 minutes until the flesh is opaque and flakes easily with a fork.

Add unsalted butter to the pan and let it melt, then squeeze lemon wedges into the pan and stir to create a pan sauce.

Transfer fillets to serving plates skin side up, spoon the garlic-thyme-lemon butter sauce over the fish, and sprinkle with chopped parsley.