Preheat oven to 350 degrees F. Lightly oil an 9×13 baking dish or coat with nonstick spray.
In a large saucepan of 2 cups water, cook quinoa according to package instructions. Within the last 5 minutes of cooking time, add broccoli on top and steam until cooked through.
Heat 1 tablespoon olive in a large skillet over medium high heat. Add Panko and cook, stirring, until browned and toasted, about 3 minutes; set aside.
Heat remaining 1 tablespoon olive oil in the skillet. Season chicken breasts with salt and pepper, to taste. Add to skillet and cook, flipping once, until cooked through, about 3-4 minutes per side. Let cool before dicing into bite-size pieces; set aside.
Melt butter in the skillet over medium heat. Whisk in flour until lightly browned, about 1 minute. Gradually whisk in milk, and cook, whisking constantly, until slightly thickened, about 3-4 minutes. Stir in quinoa, broccoli, chicken, 1 cup cheese and Greek yogurt; season with salt and pepper, to taste.
Spread broccoli mixture into the prepared baking dish; sprinkle with remaining 1/2 cup cheese. Place into oven and bake until cheese has melted, about 5 minutes.
Serve immediately, sprinkled with Panko, if desired.