Whisk all the ingredients together to combine; set aside.
Trim the cucumber so it is 5 inches long. Cut lengthwise into 1/4-inch-thick planks. Stack the planks up and cut lengthwise into 1/4-inch-wide sticks.
Make sure all the remaining wrap ingredients are prepped and ready to assemble into the rolls and you have a clean, dry cutting board. Fill a round cake pan or pie dish with hot water. You should be able to touch it comfortably.
Immerse one wrapper in the hot water and let soak until pliable but still slightly stiff, 15 to 20 seconds. Use your hands to carefully pick up the wrapper so it doesn't collapse on itself. Gently place it on the cutting board.
Lay 4 Mandarin oranges segments in a row on the wrapper, 2 inches from the bottom. Place a heaping tablespoon of chicken on the wrapper right above the oranges and arrange it to be the same length as the oranges. Top the oranges and chicken with 3 cucumber sticks. Sprinkle about a tablespoon of the fried noodles over the cucumber. Top the cucumber with a layer of lettuce and a layer of cabbage, tearing the lettuce and cabbage with your hands as needed so they just cover the other ingredients.
Fold the sides of the wrapper over toward the center. Roll the bottom of the wrapper tightly over the filling and continue rolling up until completely closed. Place seam-side down on a large plate or baking sheet and cover with a damp paper towel.
Continue with softening and filling the remaining wrappers, changing out the water as needed if it cools down too much. Cut the rolls in half crosswise if desired and serve with the dipping sauce.