In a bowl, whisk together vinegar, honey, soy sauce, brown sugar, garlic, ginger, and red pepper flakes. Marinade the chicken leg quarters in the refrigerator 4-24 hours.
Preheat the oven to 400ºF.
Transfer the marinade to a small saucepan. Bake chicken for 25 minutes.
While the chicken is baking, bring the marinade to a boil, then reduce the heat to a slow simmer. Cook over low heat, stirring as needed, for about 15 minutes or until thickened.
After the chicken has cooked for 25 minutes, brush on the marinade. Cook for 15 minutes more, brush on marinade again, bake an additional 5-10 minutes or until an internal temperature of 165ºF is reached. Transfer the chicken to a serving dish and sprinkle with sesame seeds and green onion.
Cook noodles per package directions and drain
In a large skillet or wok on medium-high heat, add oil. Once hot, add beans and mushrooms. Toss and sear for 4-6 minutes. Once tender and beginning to sear, transfer the cooked noodles to the skillet. Toss together the veggies, noodles and noodles seasoning packet sauce.