Heat oil in a 4-quart pot over medium heat. Add ginger and garlic. Stir and cook for 30 seconds until fragrant.
Add the tomatoes. Cook and stir until softened, about 3 minutes.
Pour in the chicken broth, oyster sauce, and light soy sauce. Stir in the tomato paste and sugar. Bring the mixture to a boil.
Add the beef slices. Cook for 3 to 4 minutes, skimming off and discarding any foam until the soup is clear.
Add the enoki mushrooms and cook until softened, about 1 to 2 minutes.
Turn off the heat. Stir in the white pepper and green onions, and adjust the seasoning with a pinch of salt if desired. Serve hot as a main or a side dish.