Rinse the quinoa well. Put the quinoa and 3 cups water in a large saucepan over high heat and bring to a boil. Reduce the heat so the water simmers and cook until the quinoa is tender, about 15 minutes. Drain and spread on a rimmed baking sheet to cool.
Heat 3 tablespoons of the vegetable oil in a large nonstick skillet or wok over medium-high heat. Add the eggplant and cook until it begins to brown, 2 to 3 minutes. Add 2 more tablespoons of the vegetable oil to the skillet and stir in the garlic, carrot, ginger and scallion whites. Cook until the vegetables begin to soften, about 1 minute. Stir in the soy sauce, sesame oil and fish sauce. Add the shrimp and cook, stirring, until they begin to change color, about 2 minutes. Add the cooled quinoa. Continue to cook, stirring, until the shrimp are fully cooked, about 3 minutes. Stir in the scallion greens. Divide the mixture among 4 bowls.
Wipe out the skillet and heat the remaining 1 tablespoon vegetable oil over medium-high heat. Crack the eggs into the skillet, sprinkle with a little salt and a few turns of pepper and cook until set, about 3 minutes.
Top each bowl with a fried egg.
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