Bring a large pot of water to a boil. Remove from heat, add noodles, and cover.

Peel and discard tough outer leaves from lemongrass. Roughly chop pale inner stalk and transfer to a food processor; pulse until minced, 1 to 2 minutes.

Bring chicken broth to a boil in another large pot. Add lemongrass, chicken, mushrooms, red chile pepper. Cook for 5 minutes. Reduce heat to medium; stir in coconut milk, ginger, fish sauce, and garlic. Simmer soup gently until an instant-read thermometer inserted into the chicken reads at least 165 degrees F (74 degrees C), 10 to 15 minutes.

Transfer chicken to a cutting board; slice thinly and return to the pot. Stir in lime juice and sugar.

Drain noodles and divide among serving bowls. Ladle soup on top. Garnish with a lime wedge, green onions, and basil.