Preheat the oven to 425 degrees F. Soak the fish in a bowl with the milk and 2 cups ice, 15 minutes. Meanwhile, mix the flour with 1/2 teaspoon each salt and pepper in a shallow dish. Lightly beat the eggs in another dish. Pulse the bread, parsley, lemon zest and 1/2 teaspoon salt in a food processor until crumbs form, then transfer to a third dish. One at a time, remove the fillets from the milk and dredge in the flour, shaking off the excess. Dip in the eggs, then coat with the breadcrumbs, gently pressing to coat both sides.
Transfer to a large plate. Line a baking sheet with foil.
Heat 2 tablespoons each olive oil and butter in a large skillet over medium-high heat.
Add 2 fillets and cook until golden, about 3 minutes per side.
Transfer to the prepared baking sheet. Repeat with the remaining 2 fillets, adding 2 more tablespoons each olive oil and butter.
Transfer the fish to the oven and bake until cooked through, about 8 minutes. Toss the arugula with the remaining 1 to 2 tablespoons olive oil, and salt and pepper to taste.
