Preheat the oven to 350°F (175°C). Grease a 9×13-inch baking dish with butter.
In a large skillet over medium heat, cook the breakfast sausage, breaking it apart with a spoon, until browned and cooked through, about 7-8 minutes. Remove the sausage with a slotted spoon and set aside. Drain excess fat from the skillet.
In the same skillet, add the butter, chopped onion, and chopped green bell pepper. Sauté until softened, about 4-5 minutes. Remove from heat.
In a large mixing bowl, whisk together the eggs, whole milk, salt, black pepper, and dried thyme until well combined.
Add the cubed bread, cooked sausage, sautéed onion and bell pepper mixture, and shredded cheddar cheese to the egg mixture. Gently fold to combine all ingredients evenly.
Pour the mixture into the prepared baking dish, spreading it out evenly.
Bake uncovered in the preheated oven for 35-40 minutes, or until the casserole is set and the top is golden brown.
Remove from oven and let cool for 5 minutes before serving.