Heat the oil over medium-high heat in a large (at least 8-quart), heavy-duty pot or Dutch oven. Add the carrot, ginger, Thai bird chile, and 1/2 Tbs. salt. Cook, stirring occasionally, until the vegetables are softened and lightly browned in places, 4 to 6 minutes.
Add the broth and bring to a boil. Reduce to a simmer and cook, stirring, until the vegetables are completely tender, 20 to 30 minutes
Stir in the reserved shredded chicken, ramen noodles, bok choy, and mushrooms. Simmer, stirring occasionally, until all are tender and the flavors meld, 10 to 15 minutes.
Stir in 1 Tbs. of the soy sauce, 1 tsp. of the vinegar, 1 tsp. of the sugar, and 1/2 tsp. of the sesame oil. Taste, and add more soy sauce, vinegar, sugar, and sesame oil, a little at a time, until the soup has a nicely balanced flavor. Season with salt and pepper to taste. Garnish each bowl with a 1 Tbs. of the scallions, if you like.