Preheat oven to 350°F. Grease an 8-by-8-inch glass or metal baking dish with butter; set aside.
Put quinoa into a fine-mesh strainer and rinse under cold running water until water runs clear; drain well.
In a large bowl, whisk together eggs, milk, garlic, thyme, salt, pepper, and quinoa. Stir in spinach, then pour mixture into prepared dish.
Cover tightly with foil then jiggle dish gently from side to side so that quinoa settles on the bottom in an even layer. Bake until just set, about 45 minutes.
Remove foil and sprinkle top evenly with cheese. Return to oven and bake, uncovered, until golden brown and crisp, 10 to 15 minutes more.
Set aside to let cool briefly, then slice and serve.