For the puttanesca: To the bowl of a food processor, add the peppers, garlic, olives, capers, tomato paste, anchovy paste and salt. Process until smooth, scraping down the sides of the bowl with a rubber spatula as needed. With the motor running, add 1 tablespoon water and the olive oil in a steady stream until emulsified and smooth. Pour the sauce into an airtight container and store in the refrigerator until ready to serve.
For the branzino: Dry the branzino fillets well with paper towels. Score the skin of each fillet three times with a sharp knife. Sprinkle with the salt.
Heat 2 tablespoons of the olive oil in a large nonstick skillet over medium-high heat. Lay 4 fillets, skin-side down, in the pan; press gently to ensure full contact with the pan. Cook until the skin is golden brown and the flesh is opaque at the edges, 2 minutes. Flip the fillets and cook for an additional 10 seconds; remove to a plate. Wipe out the pan and repeat with the remaining oil and fish.
To serve, spread some sauce on each serving plate and shingle 2 fillets on top. Drizzle with a little more sauce if desired.