Heat the olive oil in a large skillet over medium high heat. Add the sausage and cook, breaking up any clumps with a wooden spoon, until brown, about 5 minutes; transfer to a bowl with a slotted spoon. 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 a 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 Muenster cheese, followed by the sausage-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 Muenster cheese. Bake until slightly puffed and just no longer jiggly in the center, 40 to 50 minutes. Let stand 5 minutes before serving.
Copyright 2016 Television Food Network, G.P. All rights reserved.