Heat a large nonstick skillet over medium-high heat. Add onion and prosciutto, and sauté 5 minutes or until onion begins to brown. Add water; cover, reduce heat to low, and simmer 30 minutes. Uncover and simmer 30 minutes or until liquid almost evaporates. Cool.
Combine the milk, mustard, pepper, and egg substitute in a large bowl, and stir with a whisk until mixture is well-blended. Stir in the onion mixture. Add bread, tossing gently to coat.
Arrange half of the bread mixture in a single layer in an 11 x 7-inch baking dish. Sprinkle with 1/2 cup cheese, and top with remaining bread mixture. Cover strata, and chill 8 hours or 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 set.