1. Lightly grease a 2-quart casserole, or a 9" x 13" pan. Blanch the broccoli florets by boiling briefly, for about 2 minutes. Remove from the water and set aside to cool.
2. Put half the cubed bread in the prepared casserole or pan. Top with all the cooked sausage, three-quarters of the broccoli, half the cheese, and the dried onion. Top with the remaining bread and cheese. (The last of the broccoli comes later.)
3. Whisk together the eggs, milk, salt, mustard, and pepper. Pour slowly over the bread mixture, pushing the bread down into the liquid. If necessary, add a bit more milk to barely submerge the bread when you press it down. Cover and refrigerate overnight, or for at least 6 hours.
4. When you're ready to bake, take the casserole out of the refrigerator and preheat the oven to 350°F. Bake the casserole, uncovered, for 45 minutes. Place the reserved broccoli florets on top of the strata, pushing them in gently. Continue to bake till the top is golden and a knife inserted in the center comes out clean, an additional 45 minutes or so. Remove from the oven, and serve warm.