Heat the oil in a large pot or Dutch oven over medium heat. Add the garlic and ginger; sauté 1 minute until fragrant but not browned.
Add the sliced shiitake caps; cook 4 minutes, stirring, until softened and lightly browned at the edges.
Pour in the broth and soy sauce. Bring to a gentle boil, then reduce to a simmer for 10 minutes so the mushroom flavor builds into the broth.
Add the bok choy stems ; simmer 2 minutes.
Add the bok choy leaves ; simmer 3 minutes more, just until wilted and bright green. Do not add the leaves with the stems — they need far less time.
Off the heat, stir in the sesame oil. Taste and adjust salt and white pepper. Ladle into bowls and top with scallions.