Preheat the broiler with a baking sheet under the heat source. Combine 2 tablespoons oyster sauce and 1 teaspoon sesame oil in a small bowl; brush on both sides of the steak and season lightly with salt and pepper. Add the steak to the hot baking sheet and broil until browned around the edges and a thermometer inserted into the side registers 120 degrees F, 5 to 7 minutes. Transfer to a cutting board and let rest 10 minutes.
Meanwhile, heat the vegetable oil in a large nonstick skillet over medium-high heat. Add the frozen rice and quinoa and 1/4 cup water. Cover and cook, stirring occasionally, until the grains are thawed and no liquid remains, about 8 minutes.
Push the grains to one side of the skillet and add the cabbage and broccoli to the other side; season with 1/2 teaspoon salt. Cover and cook, stirring once or twice, until the vegetables are crisp-tender and charred in spots, 4 to 5 minutes. Add the scallions, ginger and garlic; stir the vegetables into the grains and cook until the scallions are softened, about 2 minutes. Stir in the remaining 2 tablespoons oyster sauce and 1 teaspoon sesame oil.
Thinly slice the steak against the grain. Divide the grain mixture among bowls. Top with the steak; drizzle with more oyster sauce.