Heat the olive oil in a large heavy pot over medium heat. Add the sliced onion and celery and cook for about 5 minutes until they begin to soften.
Add the mushrooms and cook for another 3 minutes until they soften and release moisture.
Stir in the crushed garlic, minced ginger, and chicken legs; cook for a few minutes until the chicken begins to brown slightly.
Add the chopped carrots, miso paste, a pinch of salt and pepper, and pour in 1.5 liters of water. Bring to a boil, then reduce heat to simmer, cover, and cook for 20–30 minutes until the chicken is fully cooked through.
Remove the chicken legs, shred the meat, and discard the bones. Return shredded meat to the pot.
Add the broccoli and whole-wheat noodles to the pot. Simmer 4–5 minutes more until noodles soften and broccoli is tender but still bright.
Taste and adjust seasoning with salt or pepper. Ladle into bowls and top with sliced spring onions and red chilli if desired. Serve hot.