Preheat the oven to 375 degrees F. Coat a 9- by 13-inch baking dish with cooking spray.
Whisk together the yogurt, milk, sour cream and flour in a large bowl. Stir in the hash browns, 1/2 cup of the Cheddar, 2 teaspoons salt and 1/2 teaspoon pepper. Set aside.
Melt the butter in a large skillet over medium-high heat. Add the onions and cook, stirring often, until soft and lightly browned, about 10 minutes. Add the mushrooms and cook, stirring often, until the mushrooms are wilted and the onions begin to brown, about 10 minutes more. If the pan gets too hot, add a splash or two of water as needed. Stir in the garlic and cook for 1 minute. Stir the vegetables into the hash brown mixture to combine.
Spread the mixture evenly in the prepared dish. Sprinkle with the remaining 1/2 cup cheese, the breadcrumbs and nutmeg if using. Bake until a deep golden brown, about 45 minutes.
Sprinkle with parsley and serve.