Heat 1-1/2 Tbs. of the oil in a 5- to 6-quart Dutch oven over medium-high heat until shimmering hot. Season the chicken with 1/2 tsp. each salt and pepper, and cook without disturbing until it’s browned and releases easily from the bottom of the pot, about 2 minutes. Flip and cook until the second side is browned and almost firm to the touch (just short of cooked through), 1 to 2 minutes more. Transfer the chicken to a cutting board to cool.Add the remaining 1 Tbs. oil and the shallots to the pot. Sprinkle with 1/4 tsp. salt, reduce the heat to medium and cook, stirring, until the shallots start soften, about 2 minutes. Add the lemongrass, ginger, and brown sugar and cook, stirring, until the ginger and lemongrass sizzle and become fragrant, about 1 minute. Add the chicken broth, scraping up any browned bits from the bottom of the pot, and raise the heat to medium high. Bring the broth to a boil and then reduce to a simmer. Add the mushrooms, and cook, stirring occasionally, until tender, 5 to 7 minutes. Meanwhile, bring a medium pot of well-salted water to a boil and cook the noodles, stirring, until just tender, about 3 minutes. Transfer to a colander and run under cold water to cool slightly. Drain well. Use your fingers or the tines of a fork to shred the chicken. Add the chicken and noodles to the broth and cook until the noodles are completely tender and the chicken is cooked through, about 2 minutes. Discard the lemongrass. Stir in the chiles, torn basil, lime juice, and soy sauce; season with more soy to taste. Divide the noodles among 4 large, deep bowls. Ladle the soup over the noodles, and garnish with the basil sprigs and scallions, carrot, and cilantro, if using. Serve with the lime wedges for squeezing.