Combine the first 3 ingredients in a small saucepan. Bring to a boil, and cook until reduced to 1 cup (about 8 minutes).
Remove from heat.
Combine egg whites and egg in a medium bowl, beating with a whisk until blended. Stir in milks, 1/3 cup brown sugar, vanilla, cinnamon, and nutmeg. Arrange half of bread slices, slightly overlapping, in the bottom of an 8-inch baking pan coated with cooking spray. Spoon fruit mixture evenly over bread. Arrange remaining bread over the fruit mixture.
Pour egg mixture over bread.
Sprinkle top with 4 teaspoons brown sugar and butter.
Place pan in a 13 x 9-inch baking pan; add hot water to larger pan to a depth of 1 inch.
Bake at 350 for 45 minutes or until a knife inserted in center comes out clean.