For quinoa cakes:
1 1/2 cups water
1 cup quinoa
1 large egg, lightly beaten
dried parsley flakes or oregano to taste
4 to 5 Tablespoon olive oil, divided
1 1/2 pounds eggplant, cut into 1/2 inch cubes
1 small onion, finely chopped
2 teaspoons finely chopped garlic
1/2 teaspoon dried oregano
3 Tablespoons olive oil
1 cup grape or cherry tomatoes, halved
1/2 cup drained bottled roasted red peppers, rinsed and chopped
3/4 cup water
1 Tablespoon chopped flat-leaf parsley
1/4 pound smoked mozzarella, diced (1 cup)
Make quinoa cakes:
Bring water and 1/2 teaspoon salt to a boil in a heavy medium saucepan.
Meanwhile, wash quinoa in 3 changes of water in a bowl, then drain well in a fine-mesh sieve.
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.
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.
Make Ragu while quinoa chills:
Toss eggplant with 1 teaspoon salt in a colander and drain 30 minutes.
Squeeze handfuls of eggplant to extract liquid, then pat dry.
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.
Cook quinoa cakes:
Heat 2 tablespoon 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.
To serve, 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.
Quinoa cakes can be formed 1 day ahead and chilled, covered.
Eggplant-tomato ragu, withour parsley and mozzarella, can be made 1 day ahead and chilled, covered.