Preheat the oven to 375°F. Lightly grease a 9x13-inch baking dish with a little butter or nonstick spray.
Place the thawed hash browns in a clean tea towel or several layers of paper towels and squeeze firmly to remove as much liquid as possible — this keeps the casserole from becoming watery.
Heat a large skillet over medium and add the turkey sausage, breaking it up with a spatula. Cook until browned and no longer pink, about 6–8 minutes. Transfer the cooked sausage to a paper towel-lined plate to drain.
In the same skillet, add the chopped onion and sauté in the remaining fat (add a teaspoon of butter if the pan is dry) until soft and translucent, 4–5 minutes. Stir in the garlic and cook 30 seconds more. Remove from heat.
Toss the drained hash browns with the melted butter and half of the shredded cheddar in a large bowl. Season lightly with 1/2 teaspoon of the salt and a few grinds of pepper.
Press half of the seasoned hash brown mixture into the bottom of the prepared baking dish to form an even base. Spread the sautéed onion and browned turkey sausage evenly over that layer, then top with the remaining hash browns.
Whisk together the eggs, milk, remaining salt and pepper, and the smoked paprika in a medium bowl until smooth. Pour this custard evenly over the layered potatoes and sausage. Sprinkle the remaining cheddar over the top.
Cover the dish tightly with foil and bake for 35–40 minutes. Remove the foil and continue baking an additional 12–15 minutes, or until the center is set and the top is golden and bubbly.
Let the casserole rest for 10 minutes before slicing so it firms up and is easier to serve. Garnish with chopped parsley or chives and serve warm.