Combine the sugar, vinegar, and salt in a small saucepan to create the brine. Heat the mixture over high heat, stirring frequently until the salt and sugar are dissolved. Add ⅓ cup of water and set aside to cool.
Slice the cucumber, shred the carrot, and slice the onion. Pack the vegetables into a jar or other glass bowl, layering with the sesame seeds.
Pour the brine over the vegetables, making sure all of the vegetables are covered. Top the jar with the optional chili oil. Place a lid on the jar.
Invert the jar a couple of times to mix everything well or stir the ingredients in the bowl. Cover the bowl with plastic wrap and chill the salad for about 30 minutes.
Transfer to a serving dish, straining the liquid as desired, and serve chilled.
Nutrition facts are sometimes provided below and are calculated using an online calculator. With specific brands of ingredients and additions, omissions, or substitutions the nutrition facts may change. We encourage you to use your own nutrition facts calculator to obtain the most accurate nutrition facts for your meal.