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Add ginger and garlic to a mixing bowl, and pour over the 1/4 cup rice vinegar. Let sit for 30 seconds, then whisk in soy sauce, salt, brown sugar, ketchup, the 1 teaspoon sesame oil, and gochujang.
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Add chicken thighs to marinade. Toss and turn until thoroughly coated. Cover and marinate in the refrigerator, at least 1 hour and up to 8 hours.
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Preheat the oven to 425 degrees F (220 degrees C). Grease a 9x13-inch casserole dish with 1 teaspoon sesame oil.
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Lay rice vermicelli in the dish, pulling the dry noodles apart as you do, to cover the bottom evenly, breaking off sections of noodle as needed.
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Evenly scatter over mushrooms, peppers, and about half the green onions. Top with marinated chicken thighs, smooth side up. Spoon excess marinade on top of chicken.
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Rinse marinade bowl with chicken broth and pour into the bottom of dish, without rinsing marinade off chicken. Top with a few tablespoons of green onions.
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Bake in the preheated oven until chicken is no longer pink at the center and juices run clear, 30 to 35 minutes. An instant-read thermometer inserted near the center should read 165 degrees F (74 degrees C).
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Remove chicken to a plate. Toss noodles and veggies in pan until evenly combined. Add the remaining 1/4 cup rice vinegar and 2 tablespoons sesame oil; toss one last time. Taste for seasoning.
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Serve chicken over noodles; sprinkle with sesame seeds and more green onions.