1In a large saucepan over medium-high heat, warm 2 tablespoons of the oil. Add the yellow onion, sprinkle with salt, and sear, without stirring, until browned, about 5 minutes. Stir in the garlic, ginger, and the broth and deglaze the pan, stirring and scraping up any browned bits from the bottom. Reduce the heat to medium-low, partially cover the pan and simmer until fragrant, about 30 minutes.
2Strain the broth through a fine-mesh sieve into a bowl and discard the solids.
3In the saucepan over medium heat, add the remaining 1 tablespoon of oil. Add the mushrooms and cook, stirring, until tender. Add the chicken and stir to combine. Add the broth and season to taste with soy sauce and sesame and/or chile oil. Bring to a simmer, then reduce the heat to low, cover partially, and simmer until warmed through.
4Meanwhile, put the eggs, if using, into boiling water and simmer for 5–6 minutes. Remove the eggs from the water, let cool until they can be handled, and peel them. Cook the ramen noodles according to the package directions.
5Divide the noodles evenly among individual bowls. Ladle the broth and chicken over the noodles, dividing them evenly, then sprinkle with the green onions. If desired, top each bowl with an optional halved soft-boiled egg and serve right away.