Preheat the oven to 350˚F (177˚C). Grease a 9-inch round baking dish with ½ tablespoon unsalted butter.
In a medium mixing bowl, whisk together pumpkin puree and eggs until combined. Add heavy cream, milk, sugar, salt, cinnamon, nutmeg and vanilla extract. Whisk to blend.
Cut the croissants into half from the side into top layers and bottom layers, using a sharp serrated knife. Set aside the top of the croissants. For the bottom of the croissants, cut into ½-inch cubes.
Place the cubes in the baking dish in a single layer. Sprinkle ¼ cup chocolate chips all over. Pour half of the custard on top. Lay the top layer of croissants over the pudding. Press down gently. Pour the rest of the custard over. Press down again. Add the rest of the chocolate chips and sprinkle with turibinado sugar.
Cover with a piece of aluminum foil loosely, and bake in the oven for 35 minutes. Remove the foil and continue to bake for 5 - 10 minutes, or until the pudding puffs up and the custard is set. Remove from oven and let cool for 10 minutes.
In the meantime, make the sauce. In a small heavy saucepan over medium heat, melt the butter. Add sugar, heavy cream and salt. Stir to blend. Bring to a gentle boil and cook over medium-low heat for about 5 minutes, stirring occasionally. Remove from heat and stir in vanilla. Taste (could be very hot) and adjust with more salt if needed. The sauce will thicken more as it cools.
To serve, scoop a good portion of pudding onto a serving plate. Top with ice cream and drizzle about a teaspoon of sauce over. Serve warm.