Heat the oven to 350°F (175°C) and grease a 9-by-13-inch baking dish. Blanch the asparagus in boiling water for 2–3 minutes, until bright green and barely tender, then drain and cool briefly.

Melt the butter or warm the olive oil in a skillet over medium heat. Sauté the chopped onion for 4–5 minutes, stirring occasionally, until softened but not deeply browned.

Spread the cubed day-old bread evenly in the prepared baking dish. Scatter the asparagus, softened onion, and about 1 cup of the cheddar over the bread.

Whisk the eggs, milk or half-and-half, salt, and black pepper in a bowl until well combined. Pour the custard slowly over the bread and vegetables, aiming to moisten the casserole evenly.

Sprinkle the remaining cheddar over the top, followed by the Parmesan if using. Let the assembled casserole rest for 10–15 minutes before baking so the bread can absorb the custard.

Bake the casserole uncovered for 35–45 minutes, until the center is set, puffed, and no longer wet. A knife inserted near the center should come out mostly clean.

Let the baked casserole stand for 5–10 minutes before cutting. Serve warm in generous squares with a crisp salad, fresh fruit, or your favorite breakfast-style side.