Cook the butter and 2 tablespoons extra-virgin olive oil in a heavy medium saucepan over low to medium heat just until the butter is melted, stirring frequently. Add the anchovies and stir until the anchovies dissolve, about 2 minutes. Add the garlic and cook just until fragrant, about 30 seconds. Remove from the heat. Stir in the orange juice, basil, and lemon and orange zests. Season the sauce, to taste, with salt.
The bagna cauda sauce can be made 1 day ahead. Cool, then cover and refrigerate. Rewarm before using.
Meanwhile, preheat oven to 200 degrees F.
Sprinkle the fish with salt and pepper and brush both sides of the fish with remaining 2 teaspoons extra-virgin olive oil. Working in 2 batches, fry the fish until just opaque in center, about 3 minutes per side. Transfer fish to platter. Cover with foil and keep warm in the oven while cooking the second batch of fish. Drizzle the sauce over and around the fish and serve.