In a small bowl, whisk together remaining soy sauce, vinegar, soy sauce, brown sugar, and 1 tsp. sesame oil to make a sauce. Arrange cooked and drained noodles in a large bowl. Set both bowls aside.
Heat 1 tsp. remaining oil in a wok or large, deep skillet over MEDIUM-HIGH heat. Add bell peppers, garlic, onions, and carrots and stir-fry until just softened, 5- 7 minutes. Add 1 Tbsp. reserved sauce and cook until fragrant, 2-3 minutes more. Transfer vegetable mixture to bowl with noodles. Add mushrooms to skillet, cook for 3-4 minutes, and again transfer to bowl with noodles.
Return wok to the heat, add remaining 1 tsp. sesame oil, add beef and stir-fry until just cooked through, 3-4 minutes. Transfer beef and its juices to the bowl with noodles and vegetables. Add spinach and remaining reserved sauce and gently toss to combine.
Transfer japchae to bowls, scatter sesame seeds over the top, and serve.
