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In a medium bowl, cover rice vermicelli in cold water and soak until softened, 5 to 10 minutes. Drain and set aside. Soak dried shiitake mushrooms in warm water until hydrated and softened, about 10 minutes, then drain and slice 1/4-inch thick.
Heat 2 tablespoons oil in a large, deep skillet over medium. Add beaten eggs, swirl to create an even layer, and cook until eggs are set and cooked through, about 1 minute. Transfer the cooked eggs onto a cutting board. Let cool slightly, then cut into matchsticks. Set aside.
Return the skillet to the stove and raise heat to medium-high. Add remaining 3 tablespoons vegetable oil, the shallots and shiitake mushrooms. Season with salt. Cook, stirring frequently, until the edges of shallots begin to brown, 3 to 4 minutes.
Add carrot, season with salt, and cook, stirring frequently, until softened but still crisp, 1 to 2 minutes. Add cabbage, season with salt, and cook, stirring frequently, until slightly wilted, 1 to 2 minutes.
Add tofu, soy sauce, drained rice vermicelli and 1 cup water, and cook, stirring frequently, until the noodles absorb the water, 5 to 6 minutes. Season with salt and white pepper, and stir in reserved eggs to combine. Serve topped with cilantro and chile oil.