Prepare the Bread Pudding: Whisk together egg yolks, milk, vanilla extract, cinnamon, salt, 2 cups heavy cream, and 2/3 cup granulated sugar in a large bowl until thoroughly combined. Add bread pieces, and toss to coat pieces evenly. Cover and chill 8 to 12 hours to allow bread to absorb the liquid.

Preheat oven to 350°F. Gently rub peaches under running water to remove fuzz, and chop peaches. (You will have about 6 1/2 cups.) Beat egg whites with an electric mixer on high speed until soft peaks form. Remove soaked bread from refrigerator; stir in 4 tablespoons salted butter pieces and chopped peaches. Fold in whipped egg whites until combined. Transfer mixture to a well-greased (with butter) 13- x 9-inch baking dish or a large cast-iron skillet. Cover loosely with aluminum foil, and bake in preheated oven until it puffs in the center, about 45 minutes. Remove foil, and bake until the top is lightly browned, 25 to 30 minutes.

Prepare the Bourbon Caramel: Melt 2 tablespoons unsalted butter in a medium saucepan over medium-high; stir in water and 1 cup granulated sugar. Cook, without stirring, until mixture is a medium shade of brown, 4 to 6 minutes. Carefully whisk in 1 cup heavy cream. (Mixture will bubble up as the cream is added.) Remove from heat, and whisk in bourbon. Serve Bourbon Caramel warm with the Bread Pudding.