Preheat oven to 425° F. Line a baking sheet with parchment paper.
Toss butternut squash with 1 tablespoon olive oil, sprinkle with salt and pepper, and place in an even layer on one-half the prepared baking sheet. Toss the broccolini with 1 tablespoon olive oil, sprinkle with salt and pepper, and place in an even layer on the other half of the baking sheet. Roast until the vegetables are tender and start to brown, 15 -18 minutes.
Heat 1 tablespoon olive oil in a large saucepan over medium-high heat. Add the mushrooms and cook until tender and browned, 5 – 7 minutes. Remove from pan and set aside.
Add 1 more tablespoon of olive oil to the saucepan, reserve a handful of the green onions and add the rest to the skillet with the garlic and chili garlic sauce, cook, stirring for 1 minute. Pour in about 1 cup of the broth, add the tamari and miso, and whisk well to dissolve the miso. Add the remaining broth and bring to a boil. Add the noodles and cook according to the package directions.
Divide the noodles and broth between 4 bowls. Top with the mushrooms, reserved green onion, butternut squash, broccolini, lime wedges, and furikake.