Preheat oven to 425°. In a medium bowl, mix butter with capers, lemon juice and parsley and season with salt. Hold at room temperature.

Season branzini cavities with salt and stuff 2 lemon rounds and 1 rosemary sprig in each. Season fish with salt.

In a large, nonstick, ovenproof skillet, heat 2 tablespoons olive oil until shimmering. Add 2 branzini and cook over high heat until branzini skin is browned and crisp, about 3 minutes per side. Repeat with remaining 2.

Transfer fish to a large rimmed baking sheet. Roast fish in the oven for about 10 minutes, until just cooked through.

Serve whole, passing caper butter at the table.