Coat a 6-quart slow cooker generously with cooking spray.
Place the eggs, milk, sour cream, cream cheese, salt, pepper, and mustard in a large bowl and whisk until smooth and thoroughly combined; set aside.
Place the sausage in a large skillet over medium-high heat and cook, breaking up the meat into small pieces with a wooden spoon, until browned and cooked through, 7 to 8 minutes. Using a slotted spoon, transfer the sausage to the slow cooker.
Return the skillet to medium heat and add the butter. Add the onion and bell pepper and sauté until soft and golden-brown, scraping the bottom of the skillet with a wooden spoon to loosen any browned bits, 5 to 6 minutes total. Transfer the onion and bell pepper to the slow cooker.
Add the frozen hash brown potatoes and 1 cup of the cheddar cheese and stir to combine. Pour in the egg mixture and stir until completely incorporated, then spread into an even layer.
Cover and cook until the edges brown and an instant-read thermometer inserted into the center of the casserole registers 160°F, 7 to 8 hours on the LOW setting or 2 to 3 hours on the HIGH setting. Sprinkle the remaining 1 cup cheddar cheese over the top of the casserole when there is 30 minutes remaining. Cover and continue to cook until the cheese melts completely.
