Heat 1 tablespoon neutral oil in a large pot. Add the leeks and cook for 3 minutes.
Sauté squash:
Add the butternut squash to the leeks and season with salt and pepper. Cook for 5–6 minutes until it just begins to brown around the edges.
Simmer the broth:
Add the tamari, 2 tablespoons mirin, sesame oil, maple syrup, and chili oil and toss to coat. Add the minced scallion whites. Pour in the water or broth and bring to a boil. Reduce heat and simmer for 30 minutes until the squash is very tender. Taste and season.
Blend the broth (optional):
If you'd like a smooth broth, use an immersion blender to blend the broth until smooth and creamy. Pour in the coconut milk and shichimi togarashi and simmer for 10 minutes—taste and season to preferences.
Cook the noodles:
While the broth simmers, bring a pot of water to a boil and cook the noodles according to package instructions. Drain and set aside.
Fry the mushrooms and Brussels sprouts:
Heat the remaining 2 teaspoons neutral oil in a skillet over medium-high heat.
Add the mushrooms to the skillet and cook for 4–6 minutes (depending on variety) until they brown and crisp up. Season with a pinch of salt.
Add the Brussels sprouts and continue cooking until they char and begin to soften, about 8–10 minutes depending on the size. Adjust the heat as necessary to prevent them from burning. Add a sprinkle of salt.
Add the remaining 1 tablespoon mirin to deglaze the skillet. Toss to coat. Cook for 1–2 minutes. Remove from heat.
To serve:
Divide the cooked noodles between bowls and ladle the broth over the noodles. Pile the charred vegetables on top and garnish with more shichimi togarashi and a drizzle of chili oil. Finish with a sprinkle of minced scallion greens.