Special equipment: a deep-fry thermometer
For the dip: In a medium bowl, mix the sour cream, mint, lemon juice, sugar, salt and pepper together until blended. Cover and chill for at least 30 minutes and up to 1 day.
For the fries: In a large heavy-bottomed saucepan, heat the oil over medium heat until a deep-frying thermometer inserted in the oil reaches 350 degrees F. (If you don¿t have a thermometer a cube of bread will brown in about 3 minutes.)
Place the egg whites in a medium bowl. Place the panko breadcrumbs in another medium bowl. Working in batches, drop the carrot sticks into the egg whites and turn to coat. Remove the carrot sticks, allowing any excess egg white to drip back into the bowl. Drop the carrot sticks in the panko; toss gently to coat.
Working in batches, fry the carrot sticks until deep brown, 3 to 4 minutes, turning and separating with a wooden spoon and adjusting the temperature to maintain 350 degrees F. Drain on paper towels.
Place a bowl of the dip in the center of a platter. Sprinkle the fries with salt and pepper. Arrange the fries around the dip, and serve.