Heat the oven and prepare the pan: Preheat to 375°F (190°C). Grease a 9×13-inch baking dish with nonstick spray or butter.
Build the potato layer: Spread hash browns in an even layer. Season lightly with a pinch of salt and pepper. Press gently so they sit flat in the pan.
Add fillings: Evenly scatter the ham (or other cooked protein), green onions, and half of the shredded cheese over the potatoes. If using vegetables, distribute them in a thin, even layer so the custard can flow around them.
Make the egg mixture: In a large bowl, whisk the eggs, milk (or half-and-half), yogurt (if using), salt, pepper, garlic powder, dry mustard, and paprika until smooth and well combined.
Assemble: Slowly pour the egg mixture over the layered ingredients, tilting the pan or using a spatula to encourage the custard into corners. Sprinkle the remaining cheese over the top.
Bake: Place on the center rack and bake 40–50 minutes, until the top is golden and the center is set. A thin knife inserted in the middle should come out without wet egg. If the top browns too quickly, tent loosely with foil for the last 10 minutes.
Rest and serve: Let the casserole stand 10–15 minutes before slicing. Cut into squares and serve warm with chives, hot sauce, salsa, or a simple salad.