Preheat your oven to 350°F (175°C) and lightly grease a 9x13-inch baking dish with non-stick cooking spray.
Cook the sausage in a large skillet over medium-high heat, breaking it up with a spoon, until it is browned and no longer pink, which should take about 8-10 minutes. Transfer the cooked sausage to a paper towel-lined plate to drain, leaving 1 tablespoon of drippings in the skillet.
Add the chopped onion and bell pepper to the skillet with the drippings and sauté over medium heat until the vegetables are softened, about 5 minutes. Remove the skillet from the heat.
In a large mixing bowl, vigorously whisk together the eggs, milk, dry mustard, salt, and black pepper until the mixture is smooth and fully combined.
Spread the cubed bread evenly in the bottom of the prepared baking dish. Top with the cooked sausage, sautéed vegetables, and shredded cheddar cheese. Pour the egg mixture evenly over everything, pressing down lightly with a spatula to ensure the bread soaks up the liquid.
Bake for 45 to 55 minutes, or until the top is golden brown and the center is fully set with no liquid jiggle. Let the casserole stand for 10 minutes before slicing. Garnish with chopped chives and serve.