Thinly slice white and light green parts of scallions (½ cup); set aside. Thinly slice dark green parts of scallions to equal ½ cup; set aside separately.
Melt butter in a large pot over medium-high. Add white and light green parts of scallions, carrots, celery, black pepper, and ½ teaspoon of the salt. Cook, stirring occasionally, until softened, about 5 minutes.
Add flour, miso, and garlic to vegetable mixture in pot. Cook, stirring constantly, until mixture is fragrant and flour turns light brown, about 1 minute.
Add broth to pot, stirring constantly and scraping bottom of pot using a wooden spoon to release browned bits. Bring to a boil over medium-high; add egg noodles, and cook, stirring occasionally, until tender and just cooked through, about 8 minutes.
Reduce heat to medium-low. Add chicken, ¼ cup dark green parts of scallions, and remaining ¾ teaspoon salt. Cook, stirring often, until chicken is heated through, about 1 minute. Remove from heat; stir in cream. Divide soup evenly among 6 bowls; top with remaining ¼ cup dark green parts of scallions. Garnish with additional black pepper.
