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Position racks in the upper and lower thirds of your oven and preheat the oven to 400 degrees F. Generously coat 2 large rimmed baking sheets with nonstick spray. Coat a 9x13-inch or similar casserole dish with nonstick spray and set aside. (The dish you see in these photos is 8 x 10 and is very full.)
Place 2 cups broth and the quinoa in a large saucepan, then bring to a low boil. Reduce heat to low, cover, and let cook for 15 minutes. Remove from heat and let stand, covered, for 5 minutes. Fluff with a fork and set aside.
While the quinoa cooks, roast the broccoli: place the broccoli in a very large bowl. Drizzle with 2 tablespoons olive oil and sprinkle with 1/2 teaspoon salt and ¼ teaspoon black pepper. Toss to coat the florets. If any seem dry, drizzle in a bit more oil as needed. Divide the broccoli between the two baking sheets, spreading it into an even layer and ensuring that the broccoli does not overlap.
Roast the broccoli on the upper and lower thirds of the oven for 10 minutes. Remove from the oven and use a spatula to lightly toss the broccoli and spread it back into an even layer to promote even browning. Return the pans to the oven, switching their positions on the upper and lower racks. Continue baking until the broccoli is browned and the tips are becoming crisp, about 5 to 10 additional minutes. Turn the oven down to 350 degrees F.
In a Dutch oven or similar large, deep sauté pot, heat the remaining 1 tablespoon oil over medium heat. Add the onion and carrots. Sauté until slightly softened, about 5 minutes.
Add the garlic, remaining 1 teaspoon salt,and 1/4 teaspoon black pepper. Stir and let cook just until the garlic is fragrant, about 30 seconds.
Sprinkle the flour over the top. Let cook until the flour is golden and smells lightly nutty, about 30 seconds to 1 minute, stirring until no white bits remain.
Slowly pour in the remaining 1 cup broth, then the milk, adding a few splashes of liquid at a time. Whisk constantly and pause as needed to break up any lumps. Increase the heat to bring the mixture to a simmer, then reduce the heat and let gently bubble until the sauce has thickened slightly, about 5 minutes, stirring often.
Stir in the oregano and cayenne. Remove from the heat.
Slowly stir in the quinoa, then fold in the white beans and roasted broccoli. Taste and adjust seasoning as desired.
Slowly stir in the Greek yogurt, 1/2 cup mozzarella cheese, and 1/2 cup cheddar cheese.
Transfer the mixture to the prepared baking dish and spread evenly in the pan. Top with remaining cheddar and mozzarella.
Bake, uncovered, until the cheese is melted and the casserole is hot, about 15 minutes. Turn the oven to broil, then continue cooking until the cheese is lightly browned, about 3 to 4 additional minutes, watching carefully so that the cheese does not burn. Let rest 5 minutes, sprinkle with chopped herbs and serve.