Preheat the oven to 350°F (175°C). Grease a 9×13 inch baking dish with butter or non-stick spray.
In a large skillet over medium heat, cook the breakfast sausage, breaking it up with a spoon, until browned and cooked through, about 8 minutes. Drain excess fat and set aside.
In a large mixing bowl, whisk together the eggs, whole milk, salt, black pepper, and dried thyme until fully combined.
Add the cubed bread, cooked sausage, and 1 1/2 cups of shredded cheddar cheese to the egg mixture. Stir gently to combine all ingredients evenly.
Pour the mixture into the prepared baking dish, spreading it out evenly.
Sprinkle the remaining 1/2 cup of shredded cheddar cheese evenly over the top of the casserole.
Bake in the preheated oven for 40 to 45 minutes, or until the casserole is set in the center and the top is golden brown.
Remove from the oven and let the casserole cool for 5 minutes before serving.