01 - Preheat the oven to 375°F. Lightly grease a 9x13-inch baking dish with olive oil or butter.
02 - Prepare quinoa according to package directions. Fluff with a fork and set aside.
03 - Heat olive oil in a large skillet over medium heat. Sauté the diced onion until softened, 3 to 4 minutes. Add the minced garlic and cook for 1 minute. Stir in the broccoli florets and sauté until just tender, about 5 minutes for fresh or until heated through if using frozen.
04 - In a large mixing bowl, whisk together the eggs, milk, sour cream or yogurt, dried thyme, paprika, salt, and pepper until smooth. Fold in the cooked quinoa, sautéed vegetables, and 3/4 cup of the shredded cheese.
05 - Transfer the mixture into the prepared baking dish, spreading it into an even layer. Scatter the remaining cheese over the top and sprinkle with breadcrumbs if using.
06 - Bake uncovered for 30 to 35 minutes until the edges are bubbly and the top is golden brown. Let the casserole rest for 5 minutes before slicing and serving.