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. Remove 8-inch pan from water. Serve warm with Crème Anglaise.