Using a pastry brush, spread the 2 tablespoons of the butter on the bottom and sides of a 9x13 baking dish.
In the bottom of your buttered baking dish, layer ½ of your bread cubes, then 1 cup of your diced peaches, and repeat with the remaining bread and peaches.
In a large mixing bowl, whisk together the eggs, half & half, sugar, vanilla extract, cinnamon, and nutmeg until well combined.
Very carefully pour the egg mixture over the top of the bread mixture. Make sure to go slowly to allow the bread to start to absorb the egg mixture. Gently press down on the bread pieces to make sure they are all coated with the egg custard.
Melt the remaining 4 tablespoons of butter and evenly drizzle it over the top of the bread pudding mixture.
Cover tightly with aluminum foil and refrigerate the peach bread pudding for 4 hours or overnight.
Preheat oven to 350°F. Remove your peach bread pudding from the refrigerator while your oven is preheating. Keep the aluminum foil on for the first 30 minutes of baking, remove the aluminum foil and continue to bake the peach bread pudding for an additional 30 minutes or until a knife inserted into the center comes out clean.
While your peach bread pudding is baking, you can make your caramel sauce by placing your butter into a small saucepan. Melt your butter on medium heat, then add your brown and granulated sugar. Whisk your butter and sugar to incorporate it and allow it to cook for 3 minutes, or until it reaches 220*F on a candy thermometer. Tip: Make sure you are whisking constantly to prevent the butter and sugar mixture from burning. You will notice that the butter and sugar will start to boil and almost look like a thick brown foam. That is perfect!
Remove from the heat and then add the half & half, vanilla extract, and salt. Very carefully whisk the caramel sauce until it is smooth. This caramel sauce will be extremely hot and can cause nasty burns on the skin if not handled with care.
Once your peach bread pudding is done, remove it from the oven and carefully pour your caramel sauce over the top. Tip: Reserve some of the sauce to add to individual pieces when serving. Allow your peach bread pudding to rest for 15 minutes to allow some of the caramel sauce to soak into the bread before serving.