1In a large saucepan over medium-high heat, warm 2 tablespoons of the oil. Add the yellow onion, sprinkle with salt, and sear, without stirring, until browned, about 5 minutes. Stir in the leek, garlic, ginger, dried mushrooms, and 6 cups water and deglaze the pan, stirring and scraping up any browned bits from the bottom. Reduce the heat to medium-low, partially cover the pan and simmer until fragrant, about 1 hour. Strain the broth through a fine-mesh sieve into a bowl and discard the solids.
2Meanwhile, preheat the oven to 400°F. Add the sweet potatoes to a baking sheet and drizzle with a little oil; toss and spread into an even layer. Cut the top quarter of the head of garlic off and discard. Drizzle the cut side of the garlic head with a little oil and wrap with foil. Add to the sheet pan. Roast, stirring the potatoes occasionally, until tender and tinged with brown, and the garlic is very tender. Remove from the oven.
3Squeeze the garlic cloves into a blender and add 1/3 cup of the roasted sweet potatoes with 1/2 cup broth. Puree until smooth and add to the broth.
4In the saucepan over medium heat, add the remaining 1 tablespoon of oil. Add the mushrooms and cook, stirring, until tender. Add the broth and season to taste with soy sauce. Bring to a simmer, then reduce the heat to low, cover partially, and simmer until warmed through. Whisk in the miso.
5Divide the noodles evenly among individual bowls. Ladle the broth over the noodles, dividing them evenly, then top with the roasted sweet potatoes. Sprinkle with the green onions and serve.