Preheat the oven to 400°. Grease a large cast-iron skillet or casserole dish and set aside.

Toss the Brussels sprouts and shallots with 1 tablespoon olive oil and ½ teaspoon salt on a parchment lined baking sheet. Roast in the oven until lightly browned, 25 minutes. Set aside to cool.

Meanwhile, in a small lidded saucepan, bring 2 cups of water, the quinoa and ½ teaspoon salt to boil. Cook for 15 minutes, until the liquid is absorbed. Set aside uncovered to cool.

In a large mixing bowl, combine the cooled sprouts, quinoa, egg, ½ teaspoon salt and 1 cup cheese.

Transfer to the prepared dish and spread in an even layer. Top with the remaining cheese.

Bake until the cheese has begun to brown, 15 minutes. Serve warm or at room temperature.