Pulse hazelnuts in food processor until finely chopped.

Pat fish dry and season with 1/2 teaspoon salt and 1/4 teaspoon pepper.

Heat oil in a large heavy skillet over medium-high heat until it shimmers.

Saute fish, turning once, until golden and just cooked through, about 8 minutes total. Transfer to a platter.

Melt butter in skillet, then sauté hazelnuts with garlic until nuts are golden brown, 1 to 2 minutes.

Add vinegar and water, scraping up any brown bits, then remove from heat and stir in capers and parsley.

Season with salt and pepper, then spoon over fish.