Heat 2 tablespoons vegetable oil in a large soup pot over medium-high heat. Add onions, and cook until very browned. Add ginger and garlic, and cook for another 2 minutes, until fragrant.
Add broth, water, miso, mirin, 1/3 cup tamari (soy sauce) and mushroom stems. Bring to a boil, and reduce heat to a simmer. Add crushed red pepper to taste, and cook for another 30 minutes at a low, rumbling simmer.
Prepare noodles, and cook until just tender. Drain, and rinse under cold water. Toss with sesame oil and 2 tablespoons of tamari (soy sauce). Refrigerate until ready to serve.
Drain soup broth from vegetables, and refrigerate broth for 3 hours or more. Put pan back on the stove (don’t clean it).
Thickly slice mushroom caps. Heat sesame oil in the soup pot over medium-high heat, and sauté mushrooms until brown. Scrape any brown bits from the pan, and cook until liquid is gone. Add half the bok choy and half the green onions, and sauté 1 minute. Refrigerate.
To assemble soup, place noodles in the bottom of bowls. Pour cold broth over the noodles, and garnish with cooked mushrooms, remaining raw onions, bok choy, egg and beech mushrooms. Serve with Sriracha and tamari or soy sauce.