Heat sesame oil and olive oil in a medium-large saucepan over moderate heat (see notes). Add garlic and ginger, and simmer until fragrant, about 2-3 minutes. Do not brown the garlic, or else you'll get a bitter flavor.
Add the carrots and mushrooms if you're using them, and simmer until they soften, about a minute, stirring frequently.
Add the broth, Sriracha sauce, rice vinegar, and soy sauce. Stir, and bring to a simmer; let it go for about five minutes. Taste, and adjust heat and taste to your liking by adding more Sriracha and soy sauce if needed.
While the broth simmers, cook the Ramen noodles in a separate pot as per the package's instructions. (You could cook the noodles in the broth directly, but that makes for a messy transfer to a bowl. It's much easier to transfer drained cooked noodles to a bowl and spoon the broth over top.) Once the noodles are tender, drain and rinse under cool water, place into a soup bowl, and set aside.
When the soup is ready, spoon the broth over the noodles. Allow to cool. At this point, make your soft-boiled egg if you're garnishing with one, and add the rest of your toppings to serve.