Place the fish fillets on a work surface and season with salt and pepper. Top each of the fillets with enough slices of potato to completely cover the surface of the fish, overlaying the potatoes like shingles. Sprinkle the potatoes with a bit more salt and pepper, and the 1 tablespoon rosemary leaves.
Heat a large, oven-proof skillet over medium-high heat. Add the olive oil and the butter. When the butter is melted and the oil is hot add the fish, potato side down. Place the skillet in the oven and bake until the fish is cooked through, about 15 to 17 minutes.
Meanwhile, combine the extra-virgin olive oil, chives, the remaining 1 tablespoon rosemary, salt and pepper in a small bowl.
To serve, place the fish on serving plates, potato side up. Drizzle the fish and the plate with the chive oil. Serve immediately.