Bring a large pot of salted water to a boil. Cook the pasta according to package directions. Drain.
Heat oil and butter in a large skillet over medium heat. Cook mushrooms with salt and pepper, until they soften, about 4 minutes. Set aside.
Meanwhile, bring another pot of salted water to a boil. Fill 2 large bowls with ice water. Add asparagus to boiling water and cook until blanched, about 2 minutes. Using a slotted spoon, transfer to one of the ice bath bowls. Using same boiling water, blanch the dandelion greens. Follow the same steps.
Bring a third large pot of salted water to a boil and fill a large bowl with cold water. Add the eggs and cook for 5 minutes. Using a slotted spoon, transfer the eggs to the bowl of cold water for 2 minutes. Peel eggs and halve lengthwise.
Divide noodles into bowls. Top each with mushrooms, greens, asparagus radishes and eggs. Add anchovy sauce to broth, ladle over noodles and top with sesame seeds.