Peel chayote using a potato peeler. Slice thinly around the pit. Cut slices into thin matchsticks. Sprinkle 1 teaspoon salt over chayote and rub them very well. Rinse out and set aside.
Heat oil in a skillet over medium heat. Add red chili and garlic and cook until fragrant.
Add the chayote and a pinch of salt; cook for 2-3 minutes until chayote gets somewhat soft.
Add the mushroom and stir. Add water to create the steam and continue to stir.
Drizzle Korean soy sauce for soup and green onion; toss well. Season with more salt according to your taste.
Lastly sprinkle the toasted sesame seeds and serve warm or at room temperature.