Heat the oven to 400°F and line a large rimmed baking sheet with parchment. Put the egg in a medium bowl and whisk with a fork until combined and lightly colored. Add the beans and quinoa and sprinkle lightly with salt and pepper. Mash and stir with the fork until the beans are roughly mashed and the egg is evenly distributed.
When the oven is hot, scatter 1 tablespoon of the cheese in an oval shape on one half of the parchment-lined pan; repeat with another tablespoon on the other side. Transfer half of the quinoa mixture on top of one cheese oval and half on the other, pressing down to form two cutlets about 1/2 inch thick. Drizzle the olive oil over each and scatter with the remaining cheese.
Bake undisturbed until the cutlets are browned, crisp, and firm, 30 to 35 minutes. Carefully turn the cutlets and smear each top with 1/4 cup each of the sauce and the mozzarella. Return the pan to the oven and bake until the cheese is hot and bubbly, about 10 minutes. (If you want extra sauce for passing at the table, heat some extra now.) Serve the cutlets right away, garnished with the basil.
— Recipe developed by The Bittman Project for Bob’s Red Mill