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1
Bring water and 1/2 teaspoon salt to a boil in a heavy medium saucepan.
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2
Meanwhile, wash quinoa in 3 changes of water in a bowl, then drain well in a fine-mesh sieve.
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3
Stir quinoa into boiling water and return to a boil, then simmer, covered, until quinoa is dry and water is absorbed, 20 to 30 minutes. Remove from heat and let stand, covered, 5 minutes. Transfer to a large bowl and cool, stirring occasionally, 10 minutes, then stir in egg.
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4
Line a baking sheet with plastic wrap and lightly brush with oil. Lightly oil a 1-cup dry-ingredient measure. Pack enough quinoa into measure with a rubber spatula to fill it two-thirds full. (If spatula becomes sticky, dip in water.) Unmold onto baking sheet and gently pat quinoa into a 4-inch-wide patty with spatula. Make 3 more quinoa cakes, brushing measure with oil each time. Chill cakes, uncovered, at least 15 minutes.
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5
Toss eggplant with 1 teaspoon salt in a colander and drain 30 minutes. Squeeze handfuls of eggplant to extract liquid, then pat dry.
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6
Cook eggplant, onion, garlic, oregano, and 1/4 teaspoon each of salt and pepper in oil in a 12-inch heavy skillet over medium heat, covered, stirring occasionally, until softened, about 5 minutes. Stir in tomatoes, roasted peppers, and water and simmer, covered, stirring occasionally, until eggplant is very tender and mixture is thick (if dry, thin with a little water), about 10 minutes.
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7
Heat 2 tablespoons oil in a 12-inch nonstick skillet over medium heat until it shimmers. Carefully add quinoa cakes and cook, turning once carefully and adding remaining 2 to 3 tablespoons oil, until crisp and golden, 8 to 10 minutes total (pat cakes to reshape with cleaned rubber spatula while cooking if necessary). Transfer to plates.
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8
Return eggplant ragu to a simmer and stir in parsley and half of mozzarella, then simmer, stirring, until cheese just begins to soften, about 30 seconds. Spoon over quinoa cakes, then sprinkle with remaining mozzarella.
