In a medium stockpot, warm 1-2 tablespoons of olive oil over medium heat.
Add the diced shallots to the pot, stirring well. Cook for 4-5 minutes, stirring frequently, until the shallots become translucent and begin to soften.
Trim the ends off the green onions, separating the white and green parts. Finely chop the green tops for garnish, and set aside. Dice the white parts finely.
Add the chopped white parts of the green onions, minced garlic, and ginger to the shallots. Cook, stirring occasionally, for 1-2 minutes, until the garlic and ginger become fragrant.
Slowly pour the chicken stock or water into the pot and bring to a simmer. Add the star anise and soy sauce, cover, and let simmer for 10 minutes.
Remove the lid and carefully discard the star anise from the soup.
Add the sliced mushrooms, uncooked noodles, and bok choy to the pot, allowing them to simmer for 5-8 minutes, or until the noodles and bok choy are tender. Season to taste.
Serve the soup in bowls, garnishing with sesame seeds, the green parts of the green onions, and red pepper flakes if desired.