Brown the sausage. Heat 2 Tbsp olive oil in a skillet over medium-high heat. Add sausage, then cook until browned on all sides, about 5-7 minutes. Use a slotted spoon to transfer to a large mixing bowl, leaving grease in skillet.
Cook the veggies. While still hot, add the sliced onions and mushrooms. Cook 4-6 minutes, or until beginning to brown and soften, stirring occasionally. If the pan is dry, add 1-2 Tbsp more of olive oil. Stir in 1 Tbsp chopped fresh sage, then mix well. Pour this into the bowl with the sausage, then add the cubed challah bread on top.
Mix the wet ingredients. In a separate large bowl, whisk together eggs, 1 cup of heavy cream, 1 cup of milk, 1 tsp Kosher salt, and ¼ tsp black pepper.
Mix the cheeses. Place 1 cup shredded white cheddar and ½ cup grated Parmesan in a small bowl. Stir well.
Stir everything together. Pour the egg mixture over the bread and veggies along with all except for ¼ cup of the cheese, then stir until very well mixed. Reserve remaining cheese.
Assemble the strata. Grease a 9x13 casserole dish with nonstick cooking spray (or with a few tablespoons of unsalted butter). Pour the strata mixture inside, then use a spatula to spread into an even layer. Cover with plastic wrap then refrigerate overnight (or at least ~1 hour).
Bake the breakfast strata. Preheat oven to 375°F (190°C). Remove plastic wrap, sprinkle the strata with remaining cheese, then bake for 30 minutes, or until light golden brown around the edges and the top is bubbling. Serve immediately!