In a medium bowl, combine the chicken strips and marinade. Marinate for 20 minutes.
In a large pan over medium-high heat, warm up ½ tablespoon vegetable oil. Add the chicken and cook until cook through, about 3 – 4 minutes. Transfer the chicken to a small bowl and set aside.
Meanwhile, bring a large pot of water to a boil. Follow the udon package direction to cook the udon. Drain and rinse under cold water. Set aside.
Clean the pan and pat dry. In the pan over medium-high heat, add 1 tablespoon oil. Add onion and saute for 2 minutes. Add cabbage and 3 tablespoons water. Cover and cook until almost soften. Add 1/4 cup Swiss sauce. Mix well. Add cooked udon and chicken. Toss and combine. Cook for 2 more minutes. Taste and adjust with more Swiss sauce or soy sauce if needed. Transfer to serving plates. Top with green onion. Serve immediately.
Notes
If you don’t have any leftover Swiss sauce, you can make a small portion of it. In a small pot, bring 200ml water to a boil. Add 100ml dark soy sauce, 75g rock sugar, 1 green onion (cut in half), and 1 (1-inch) ginger slice. Once the sugar is melted, simmer for another 3 minutes. The sauce is ready.
Swiss sauce is on the sweet side. Add a little soy sauce if you want to add some saltiness.