Heat a large nonstick skillet over medium heat. Add onion; sauté 10 minutes or until onion begins to brown. Add water; cover, reduce heat to low, and simmer 25 minutes, stirring occasionally. Uncover and simmer 30 minutes or until liquid almost evaporates. Cool.
Combine milk, egg substitute, mustard, salt, and pepper in a large bowl, stirring with a whisk. Stir in onion and bacon. Add bread; toss gently to coat.
Arrange half of bread mixture in an 11 x 7-inch baking dish coated with cooking spray. Sprinkle with half of cheese. Top with remaining bread mixture. Cover and chill overnight.
Preheat oven to 350°.
Uncover strata. Bake at 350° for 25 minutes. Sprinkle with 1/2 cup cheese; bake an additional 20 minutes or until browned. Let stand 10 minutes before serving.