Preheat the oven to 350°F (180°C, gas mark 4). Rinse the quinoa well and set aside to drain. Heat the oil in a pan over medium heat. Add the onion and garlic and sauté until the onion begins to brown, about 8 minutes. Add the Swiss chard and broccoli and sauté for 1 minute, then turn off the heat. Cool slightly. Transfer the mixture to a lightly greased 9- by 13-inch (23- by 33-cm) casserole dish and spread evenly. Sprinkle the Cheddar cheese evenly over the top of the vegetable mixture. Set aside.

In a large bowl, combine the eggs, milk, thyme, salt, pepper, and rinsed quinoa and whisk until thoroughly combined. Pour immediately over the cheese-topped vegetable mixture. Bake the casserole in the oven until cooked through and the top is just starting to brown, about 40 minutes. Remove the casserole from the oven and sprinkle the Parmesan over the top. Return to the oven and bake until the Parmesan is melted and just starting to brown, about 5 more minutes. Let the casserole sit for at least 10 minutes before serving. Serve with a mixed lettuce or spinach salad.