In a small bowl, combine the water, soy sauce, oyster sauce, lime juice (or rice vinegar), brown sugar, and sesame oil. Whisk together until the sugar is dissolved and set aside.
In a large skillet or wok over medium heat, add the ground chicken. Cook, breaking it apart with a spatula, until it’s browned and cooked through, about 5-7 minutes.
Once the chicken is cooked, add the sliced mushrooms, bell pepper, and the white parts of the scallions to the skillet. Stir-fry for about 3-4 minutes until the vegetables are tender but still crisp.
Add the mung bean sprouts and the udon noodles to the skillet. Pour the prepared sauce over the mixture and toss everything together until the noodles are heated through and well combined, about 2-3 minutes.
Remove the skillet from heat and garnish with the green parts of the scallions. Serve hot and enjoy your delicious Chicken Yaki Udon!