In a medium-size bowl, whisk together eggs, milk, mustard, salt, cayenne and nutmeg.
In a large bowl, toss apples, bread cubes and bacon together. Coat 13 x 9 x 2-inch baking dish with nonstick cooking spray. Arrange bread mixture in baking dish. Pour egg mixture evenly over the top. Gently press bread mixture with a wooden spoon to ensure all the bread is moistened. Top with shredded cheese. Cover with nonstick foil and refrigerate overnight.
To bake, heat oven to 400°F. Remove strata from refrigerator and bake, covered, at 400°F for 30 minutes. Remove foil and bake for 10 more minutes, until top is nicely browned. Let stand 10 minutes before serving.