Put the bacon in a large skillet then cook over medium heat, stirring occasionally, until crispy, about 10 minutes. Transfer to a paper towel-lined plate with a slotted spoon. Discard all but 2 tablespoons of the bacon drippings. Add the onion to the skillet and cook over medium heat, stirring occasionally, until very soft and golden, 20 to 25 minutes. Increase the heat to medium-high and cook, stirring often, until golden brown, about 6 minutes more. Set aside.
Whisk the half-and-half, Parmesan, eggs, 1 1/2 teaspoons salt and few grinds of pepper together in a large bowl.
Layer half of the bread cubes in the prepared baking dish. Sprinkle with 1 1/2 cups of the Swiss, followed by the bacon-onion mixture. 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 Swiss. 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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