For the eggs: Bring a medium saucepan of salted water to a boil. Add the eggs, then reduce the heat to a simmer and cook 6 minutes. Drain and run under cool water until cool, then peel.
Combine the soy sauce, sake and sugar with 1 cup water in a medium bowl and stir to dissolve the sugar. Add the eggs and place plastic wrap directly on the surface to help submerge the eggs. Cover and refrigerate at least 6 hours and up to overnight.
For the soup: Heat the sesame oil in large saucepan over medium heat. Add the garlic, ginger and half of the scallions (reserve the rest for garnish) and cook, stirring often, until translucent, about 3 minutes. Add the sriracha and stir, 30 seconds. Stir in the miso, then gradually add the chicken broth and 2 cups water, whisking. Simmer, partially covered, 10 minutes. Whisk in the butter and season with salt and pepper.
Meanwhile, bring a large pot of water to a boil. Slice the marinated eggs in half.
Cook the noodles according to their package instructions. Drain and divide among 4 bowls. Divide the soup among the bowls and top with the egg halves, remaining scallions and seaweed. Serve immediately.