In a large soup pot bring the water to a boil along with the dried mushrooms, ginger, and garlic. Adjust the heat to a gentle simmer and cook for 30 minutes. Turn off the heat and let steep, 30 more minutes.
While the broth steeps: Melt the butter over high heat in a large nonstick skillet, add the sliced fresh mushrooms and cook until crisp and brown, about 10 minutes. Season the mushrooms with salt and pepper, to taste, and distribute them evenly between 4 soup bowls.
Put the eggs in a small saucepan and cover with cold water. Bring to a boil, turn off heat, cover, and let cook until almost soft boiled, 3 to 4 minutes. Immediately rinse the eggs with cold water and set aside.
Strain the broth. Reheat and season with the sherry, soy sauce, and salt to taste. Distribute noodles, enoki mushrooms, if using, and scallions evenly in each soup bowl. Crack a soft-boiled egg into each bowl (to maintain the eggs' shape, take care to peel them carefully, or use as small spoon to free them from the shell). Ladle 2 cups of hot broth into each bowl. Sprinkle gomashio or shansho pepper over the soup, if desired, and serve immediately.