Sprinkle fillets with salt and ground peppercorns. Melt 1 tablespoon butter in large nonstick skillet over medium-high heat.
Add fillets, flesh side down. Cook until browned, 2 to 3 minutes. Turn fillets over; cook until opaque in center, 2 to 3 minutes.
Transfer to plates; cover to keep warm.
Add hazelnuts to same skillet over medium heat. Toast until golden, 1 to 2 minutes.
Add 3 tablespoons butter and cook until golden, 2 to 3 minutes. Season with salt and pepper. Divide sauce among fillets; sprinkle with parsley and grated lemon peel.