Whisk together the half-and-half, Parmesan, eggs, 1 1/2 teaspoons salt and a few grinds of pepper in a large bowl.
Layer half of the bread cubes in the prepared baking dish. Sprinkle with 1 1/2 cups of the Cheddar, followed by the sun-dried tomatoes and scallions. Cover with the remaining bread cubes and pour the egg mixture evenly over the 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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