Brown the sausage: Crumble the ground breakfast sausage into a skillet over medium heat. Cook, breaking it apart with a spoon, until no pink remains, about 7 to 8 minutes.
Drain excess fat: Remove the sausage from heat and drain off excess fat using a paper towel-lined plate. This keeps the finished casserole from turning greasy.
Soften the vegetables: Return the skillet to medium heat and add the diced bell pepper and onion. Cook until the onion turns translucent, about 3 minutes, then set aside.
Whisk the egg mixture: In a large bowl, whisk the eggs, milk, ground mustard, salt and pepper together until the color turns pale yellow and slightly foamy.
Grease the baking dish: Lightly grease a 9x13-inch baking dish with butter or cooking spray so the casserole releases easily after baking.
Layer the bread cubes: Spread the cubed day-old bread evenly across the bottom of the prepared dish, pressing lightly to form an even layer.
Add sausage and cheese: Scatter the cooked sausage, bell pepper and onion mixture over the bread, then sprinkle with half of the shredded cheddar cheese.
Pour the egg mixture: Pour the whisked egg mixture evenly over the bread and sausage layer, pressing down gently so the bread soaks up the liquid.
Top with remaining cheese: Sprinkle the remaining shredded cheddar cheese evenly across the top of the casserole.
Bake until set: Bake at 350°F for 40 to 45 minutes, until the center no longer jiggles and the top turns golden brown with crisp edges.
Rest before serving: Let the casserole rest for 5 to 10 minutes so it firms up slightly, making it easier to cut into clean squares.
Garnish and serve: Sprinkle chopped chives over the top before serving warm, straight from the baking dish.