Heat oil in a 5-quart pot over medium-high heat. Cook mushrooms for 6 minutes, stirring occasionally, until softened and lightly browned.
Stir in garlic and ginger for 30 seconds. Add broth and tofu, cover and bring to a simmer.
Add bok choy stems and simmer for 2 minutes. Add ramen noodles and cook for their package time, about 3 minutes, separating the strands and adding bok choy leaves during the final minute.
Switch off the heat. Whisk miso with 1/2 cup of hot broth in a small bowl until smooth, then stir it back into the soup with soy sauce and sesame oil.
Divide noodles, tofu, mushrooms and broth among 4 bowls and serve immediately without boiling the miso broth again.