Make dressing: In a wide, 3- to 4-qt. saucepan, boil carrot juice over medium-high heat, stirring often, until reduced to 1/4 cup, 12 to 15 minutes. Pour into a small bowl and let cool. Whisk in remaining dressing ingredients.
Meanwhile, preheat oven to 350°. Toast sunflower seeds in a shallow pan until light golden, shaking pan occasionally, 4 to 8 minutes. Let cool.
Trim off membrane and curved ends of bell pepper. Cut pepper lengthwise into very thin slices.
In a large bowl, use your hands to toss and separate all vegetables until evenly mixed. Pour seeds and dressing on top; mix gently. Season with salt to taste.
Note: Raw sprouts can occasionally contain bacteria that cause foodborne illness, so children, the elderly, and those with weakened immune systems should avoid eating them.
Note: Nutritional analysis is per serving.