Butter a 9 x 13-inch baking dish and set it aside.
To a large mixing bowl, add the pumpkin, half and half, milk, brown sugar, granulated sugar, eggs, egg yolks, cinnamon, ginger, nutmeg, cloves, allspice, salt, and vanilla and whisk until well combined.
Add the cubed French bread to the bowl with the custard and use a wooden spoon to gently mix the ingredients until all of the bread is moistened. Transfer the bread pudding to the prepared baking dish, wrap it tightly in plastic wrap, and refrigerate it for at least one hour up to overnight.
Set the bread pudding out on the counter and preheat your oven to 350°F. Once the oven is hot, pour the melted butter over the bread pudding and bake it in the preheated oven for 50 to 60 minutes, or until golden brown on top and set in the center.
While the bread pudding is baking, begin making the salted caramel sauce by combining the water and sugar in a large saucepan and heating it over medium-high heat until the sugar dissolves. Do not stir the sugar and water mixture again once the sugar has dissolved! Bring the mixture to a boil and cook it until it reaches an amber color.
Lower the heat to low and whisk in the heavy cream. The sauce will bubble up a lot when the cream is added. Stir until the cream is fully incorporated. Remove the pan from the heat and stir in the butter and vanilla extract. Set the caramel sauce aside.
Once the bread pudding is set in the center, remove it from the oven and allow it to cool for about 15 minutes. Slice bread pudding into squares and serve warm with salted caramel on top. Add a scoop of vanilla ice cream for extra decadence!