Preheat the oven to 450 degrees F. Fill a large pot with 1 inch of water and set a steamer basket inside; bring to a boil. Place the beet noodles in the steamer basket, cover and cook until tender but still holding their shape, 15 to 20 minutes. Transfer to a large bowl and set aside until ready to serve. Whisk 2 tablespoons rice vinegar, the soy sauce and half each of the ginger, garlic and serrano in a small bowl; set the dressing aside.
Season the chicken on both sides with salt and pepper. Heat 2 tablespoons sesame oil in a large ovenproof skillet over medium-high heat. Add the chicken and cook until browned on both sides but not cooked through, about 6 minutes. Transfer the skillet to the oven and roast until cooked through, about 10 minutes. Transfer the chicken to a cutting board and let rest.
Meanwhile, heat the remaining 2 tablespoons sesame oil in the same skillet over medium-high heat. Add the mushrooms; cook, stirring, until browned, 8 to 10 minutes. Season with salt and pepper. Add the remaining ginger, garlic and serrano and cook, stirring, until tender, about 1 minute. Deglaze the pan with the remaining 1 tablespoon vinegar and 3 tablespoons water.
Toss the beet noodles with the ginger dressing; divide among plates and top with scallions and mint. Slice the chicken and add to the plates. Top with the mushroom mixture.