Slice both halves of the cucumber crosswise on the diagonal to make thin crescents. In a large bowl, combine the cucumbers, carrots, and minced chile and set it aside.
Whisk together the lime juice, brown sugar, and fish sauce or salt, and stir until the sugar completely dissolves. Pour the dressing over the vegetables and toss well. Add cilantro, if desired. Chill for 30 minutes to allow the flavors to meld.