Preheat oven to 350°. Spray 9x13 inch baking dish with cooking spray. Cook sausage over medium high heat until no longer pink, about 5-7 minutes; drain and set aside.
In a large bowl, whisk together eggs, milk, salt, pepper, and thyme. Stir in potatoes, sausage, and 1 cup of cheese; pour into prepared baking dish. Bake immediately or cover and refrigerate up to 24 hours.
Bake, uncovered, 55 minutes or until eggs are set. Top with cheese, bake for additional 5 minutes.