Grease a 9x13-inch baking dish with the unsalted butter.
Cook the breakfast sausage in a large skillet over medium heat, breaking it into bite-size pieces, until browned and cooked through, 6–8 minutes. Transfer to a plate, leaving about 1 tbsp fat in the pan. Add the finely chopped onion and cook until translucent and tender, 4–6 minutes; remove from heat and let cool slightly.
Spread the bread cubes evenly in the prepared baking dish. Scatter the cooked sausage and onion over the bread. Sprinkle 1.5 cups of the shredded cheddar over the top, reserving 0.5 cup for finishing.
In a large bowl, whisk the eggs, milk, dry mustard, kosher salt, and black pepper until smooth and well combined. Pour the egg mixture evenly over the casserole, pressing down lightly so the bread absorbs the custard. Cover and refrigerate for at least 30 minutes and up to overnight.
When ready to bake, preheat the oven to 350°F (175°C). Uncover the casserole, sprinkle with the reserved 0.5 cup cheddar, and bake until the top is golden, the edges are set, and the center is just set with a slight jiggle (a knife inserted near the center should come out mostly clean), 45–55 minutes. Tent loosely with foil if browning too quickly.
Let rest 10 minutes before cutting into squares. Serve warm.