Begin by heating the cooking oil in a large pot over medium heat. Once the oil is shimmering, quickly sauté the minced garlic, ginger, and scallions. Stir frequently and cook for 30 seconds to 1 minute until aromatic.
Add the beef shank slices to the pot, searing them briefly to develop a deeper flavor. Pour in the soy sauce, cooking wine, beef broth, and water, stirring to combine all the ingredients evenly.
Toss in the star anise, dried Thai chilies, and brown sugar. Let the flavors meld by bringing the mixture to a boil. Reduce the heat to low and allow it to simmer gently for 60 to 90 minutes, or until the beef becomes tender and the broth is richly infused.
While the soup is simmering, soak the dried noodles in hot water for 10 minutes, then drain and set aside.
Once the beef is tender, add the noodles into the pot and cook for 3 minutes, or until they absorb some of the broth and soften fully.
Distribute the noodles evenly into serving bowls. Add the cleaned baby bok choy directly into the soup, season with salt and pepper, then stir gently. Cover the pot with the heat turned off and allow the residual warmth to wilt the greens for about 5 minutes before serving piping hot.