In a medium soup pot, combine the vegetable broth, water, lemongrass, sliced ginger, and soy sauce and season generously with white pepper. Bring to a simmer, cover and cook over low heat until flavorful, about 20 minutes. Strain the broth into a heatproof bowl, pressing on the solids. Discard the solids.

Meanwhile, bring a medium saucepan of water to a boil. Cook the rice vermicelli until al dente, about 5 minutes. Drain and cool under running water.

Wipe out the soup pot and heat the vegetable oil. Add the eggplant and shiitake mushrooms, season with salt and white pepper and cook the vegetables over high heat, stirring occasionally, until softened and browned, about 8 minutes. Stir in the chopped ginger and garlic and cook until fragrant, about 1 minute. Add the tofu and cook, stirring gently once or twice, until lightly browned, about 4 minutes. Add the vegetable broth along with the cabbage and noodles and simmer just until the cabbage is wilted, about 2 minutes longer. Stir in the bean sprouts, cilantro and mint, and season the soup with salt and white pepper. Serve the soup in deep bowls, passing lime wedges and hot sauce at the table.