Caramelized Onion and Breakfast Sausage Strata

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1 tablespoon unsalted butter, at room temperature

1 tablespoon olive oil

12 ounces breakfast sausage, casing removed

1 large onion, thinly sliced

3 cups half-and-half

1/4 cup grated Parmesan

10 large eggs

Kosher salt and freshly ground black pepper

One 12-ounce loaf sourdough bread, crusts removed, cut into 1-inch cubes, preferably stale (about 6 cups)

2 cups shredded Muenster cheese