Generously butter a 9x13 inch baking dish. Arrange the bread slices in the dish so they are packed in a little bit tight. In a medium bowl, whisk together the 4 eggs, milk, white sugar, vanilla, and cinnamon. Pour the liquid over the bread, and let it sit for an hour in the refrigerator. Press down the top with a spatula occasionally to help the bread absorb the liquid. After about 30 minutes, sprinkle the berries over the pudding, and press them down inside a bit to distribute.
Preheat the oven to 350 degrees F (175 degrees C). Bake the bread pudding for 45 to 50 minutes in the preheated oven, or until the top is golden, and the middle is firm. While the pudding is baking, make the brown sugar sauce.
Melt the unsalted butter in a saucepan over medium heat. Stir in the brown sugar, water, pumpkin pie spice, and salt. Cook, stirring constantly for about 2 minutes. Remove from heat, and whisk in the egg. Return to the burner, and cook stirring constantly until thickened, about 1 minute. Serve hot over the hot bread pudding.