Whisk the half-and-half, Parmesan, eggs, 1 1/2 teaspoons salt and a few grinds of pepper together in a large bowl.
Bring a medium pot of salted water to a boil. Add the broccoli florets and simmer until crisp-tender, 3 to 4 minutes; drain.
Layer half of the bread cubes in the prepared baking dish. Sprinkle with 1 1/2 cups of the Cheddar, followed by the broccoli. Cover with the remaining bread cubes and pour the half-and-half mixture evenly over top. Cover with plastic wrap and refrigerate overnight. (This can also be assembled the day of; just let soak at room temperature 30 minutes to 1 hour before baking.)
On the day of baking, preheat the oven to 325 degrees F.
Let the strata sit at room temperature while the oven preheats. Remove the plastic and sprinkle with the remaining 1/2 cup Cheddar. Bake until slightly puffed and just no longer jiggly in the center, 40 to 50 minutes. Let stand 5 minutes before serving.
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