Butter a large 6-qt. baking dish. Place bread cubes in a large bowl. In a saute' pan over medium heat, warm olive oil, add sausage and cook, turning occasionally, until browned on both sides (3-5 mins.). Drain on paper towels.
Cut sausage into thin slices; add to bread slices. Whisk together eggs and milk in a blender. Pour egg mixture over bread and sausage. Add remaining ingredients, reserving 1 cup of cheese, and stir until well blended.
Transfer to prepared baking dish, cover with plastic wrap, and refrigerate at least 4 hours or overnight. Preheat oven to 350 degrees (F). Sprinkle top of strata with 1 cup cheese. Bake until golden brown and cooked through, about 1 hour. Let stand 10 minutes before serving.