Preheat the oven to 350ºF. Spray a 9x13-inch baking dish with nonstick cooking spray.
Pour the hash browns into the casserole dish. Add 1 cup of shredded Swiss cheese, 1 cup of shredded cheddar cheese, garlic powder, and 1 teaspoon of salt to the hash browns. Mix together in the baking dish until combined. Set aside.
Heat the butter in a small saucepan over medium/high heat. When the butter is melted, reduce the heat to low, add the flour, and whisk together until a crumble forms.
Slowly whisk in the chicken broth, mixing continuously until a thick roux forms.
Mix in the remaining salt and remove the saucepan from the heat.
Sprinkle 1 cup of shredded cheddar cheese into the roux, whisking until the cheese is melted. Add the sour cream and mix until combined.
Pour the cheese sauce over the hash browns and use a spatula and combine everything together. Spread the cheesy hash browns in an even layer in the baking dish.
Bake for 30 minutes.
Remove the casserole from the oven and top with the remaining shredded Swiss and cheddar cheeses. Bake for 15 more minutes.
After 15 minutes, keep the baking dish in the oven and broil for 2-3 minutes, or until the top of the casserole is golden brown.
Let the casserole cool for 10 minutes, serve, and enjoy.