Heat the sesame oil in a medium pot over medium heat. Add the garlic, ginger and scallion whites and cook for 2 minutes, until fragrant.
Simmer the Broth
Add the stock and shiitake. Bring to a gentle boil, then reduce to a low simmer and cook uncovered for about 40 minutes, until the broth has reduced to roughly 4 cups.
Keep it at a gentle simmer. A hard boil clouds the broth and cooks off flavor.
Strain
Strain the broth, discarding the garlic, ginger and scallion whites.
Slice the shiitake caps, discard the stems, and set the caps aside to use as a topping. Return the broth to the pot and keep it hot. Don't season it yet
Cook the Eggs
Boil the eggs for 6½ minutes, then move them to ice water. Peel and halve. For more on this, see how to soft-boil eggs.
Cook the Noodles
In a separate pot of boiling water, cook the noodles according to the package, pulling them a little early since they keep cooking in the broth. Drain.
Season the Bowls
Put half the soy sauce and half the mirin in the bottom of each bowl. Ladle in about 2 cups of hot broth, stir, and taste.
Add salt and pepper if it needs it.
This is where you control the salt. Add more soy sauce a teaspoon at a time.
Assemble
Divide the noodles between the bowls. Top with the egg, shiitake, scallion greens and any other toppings.