Preheat the oven to 400 degrees . In a medium bowl, whisk together the brown sugar and soy sauce. Add the salmon and turn to coat.
Grease a 9-by-13-inch glass baking dish with the oil. Arrange the salmon in the dish, drizzle with the remaining marinade and roast until cooked through, 15 to 20 minutes. Remove from the oven and let cool for about 20 minutes.
Split the rolls in half lengthwise and spread each half with chili sauce and mayonnaise to taste. Place a piece of salmon on the bottom half of each roll and sprinkle with salt and a squeeze of lime juice. Top with some of the grated carrot and a few sprigs of cilantro. Wrap each sandwich with plastic wrap.