In a 1- to 1 1/2-quart pan, mix cardamom seed, carrot juice, 1 1/2 teaspoons vinegar, cornstarch, and sugar until well blended. Stir over high heat until mixture boils; reduce heat to low and simmer, stirring occasionally, until reduced to 1/2 cup, 7 to 9 minutes. Cover pan and remove from heat; let stand 5 to 10 minutes. Taste, and add salt and more vinegar if desired. Pour sauce through a fine strainer into a bowl.
Meanwhile, rinse fish, pat dry, and cut into 4 equal portions. Rub fish all over with olive oil and sprinkle lightly with salt. Set a 10- to 12-inch nonstick frying pan (with ovenproof handle) over high heat. When pan is hot, add fish and turn as needed to brown on both sides, 2 to 3 minutes total. Sprinkle pepper evenly over fish.
Put pan with fish in a 400° oven; bake until fish is opaque but still moist-looking in center of thickest part (cut to test), 5 to 7 minutes.
Transfer fish to rimmed plates. Spoon sauce evenly around fish and sprinkle with chives.