To a large and heavy pot over high heat, add the linguine, water, bok choy, bell pepper, carrots, onion, garlic, ginger, soy sauce, sesame oil, brown sugar, red pepper flakes, and salt.
As soon as the mixture comes to a boil, set the timer for 9 minutes and cook, tossing constantly with tongs, taking care to make sure the pasta doesn't stick to the bottom of the pot. Once the 9 minutes is up, the veggies should be tender, the pasta cooked through, and it should have a saucy consistency, with most of the liquid having evaporated.
Remove from heat and add the cabbage and vinegar, tossing until cabbage wilts, about 30 seconds.
Serve, topping individual bowls with cilantro, peanuts, scallions, and toasted sesame seeds.