Combine the heavy cream and salt in a small saucepan over low heat. Bring to a simmer and stir until the salt is dissolved.
Meanwhile, combine the water, sugar, and corn syrup in a medium saucepan over high heat, stirring until the sugar is dissolved. Bring to a boil and cook for 6 to 8 minutes, until a candy thermometer registers 350°F and the mixture is a deep amber color. Remove from the heat and let sit for 1 minute. Add the warm cream and stir until smooth. Let cool to room temperature.
Preheat the oven to 325°F. Spray a 13 by 9-inch baking dish with nonstick cooking spray.
Whisk together the eggs, brown sugar, vanilla extract, salt, milk, and cream in a large bowl to combine. Add the bread cubes and toss gently to coat all the bread in the custard. Pour half of the mixture into the prepared pan. Drizzle ¼ cup of caramel evenly over the top. Pour the remaining custard and bread into the pan and drizzle with another ¼ cup of caramel. Cover the pan tightly with aluminum foil and bake for about 1 hour, until the custard is almost set. Remove the foil and cook for another 20 to 30 minutes, until the custard is set and the top is golden brown. Remove from the oven and let cool for 30 minutes. Serve with another drizzle of caramel over the top.