In a small bowl, combine the water, soy sauce, oyster sauce, lime juice, and brown sugar to create the noodle sauce. Stir well and set aside.
Prepare the vegetables by removing the seeds from the bell pepper and slicing it into thin strips. Take the scallions, chop them into one-inch segments, and separate the whites from the greens. If your mushrooms need slicing, do so now.
Set a large wok or sauté pan over medium-high heat. Add the sesame oil once the pan is hot. This will give your vegetables a deliciously nutty flavor.
Once the oil shimmers, add the scallion whites, mushrooms, and bell pepper strips. Stir-fry the vegetables for three to four minutes until they are mostly cooked but still vibrant.
Transfer the stir-fried vegetables to a holding plate. Then, add the ground chicken to the pan. With a wooden spoon, break the chicken into small pieces as it cooks. Aim for a uniform brown color.
Once the chicken is well browned, open the packets of udon noodles and add them directly to the pan.
Pour the prepared sauce over the top of the noodles. Carefully fold the chicken and sauce over the noodles until they loosen enough to pull apart. Avoid breaking the noodles; instead, gently tug and fold to separate them.
As the noodles begin to separate, return the stir-fried vegetables to the pan. Toss everything well to combine, then add in the mung bean sprouts and the green parts of the scallions. Toss and simmer for another one to two minutes.
When everything is well combined and heated through, remove from heat. Serve the Chicken Yaki Udon immediately, garnished with extra scallions if desired.