Heat the sugar in a small non-stick skillet over medium heat. Cook, tilting the pan occasionally to help the sugar melt evenly, until dark amber (like maple syrup), about 3 minutes. Remove from the heat and carefully add 1/4 cup water; it will steam and bubble. Put the skillet over medium-low heat and cook, stirring constantly, until smooth, about 2 minutes. Add 1 tablespoon of the lime juice and the fish sauce and bring to a simmer. Cook until thick enough for a spoon to leave a trail in the skillet, about 1 minute. Remove from the heat and set aside.

In a medium bowl, combine the carrots, cilantro, and the remaining 2 tablespoons lime juice. Season to taste with more lime juice and salt. Set aside.

Heat the oil in a wok or 12-inch non-stick skillet over medium heat. Add green onions and garlic; cook, stirring occasionally, until fragrant, about 45 seconds. Add the caramel sauce and 3/4 teaspoon pepper and bring to a simmer. Add the fish and toss gently to coat. Distribute the fish in a single layer and simmer vigorously, stirring once, until cooked through, 4 to 5 minutes. Serve with the carrots.