Step 1In the bowl of a stand mixer, or in a large glass or nonreactive bowl using a hand mixer, beat the eggs. Slowly beat in the sugar, then beat in the pumpkin pie spice and vanilla extract. Beat in the mashed sweet potatoes until thoroughly combined, then beat in the butter. Add the milk, beating until all of the ingredients are completely and thoroughly incorporated. If using a stand mixer, transfer the custard base to a large glass or nonreactive bowl.
Step 2Using your hands, mix the bread cubes and raisins into the custard base. Cover and refrigerate the mixture overnight.
Step 3Shortly before baking, heat the oven to 275 degrees. Thoroughly stir the mixture, making sure there are no white spots of un-soaked bread (it is fine if the soaked bread breaks down somewhat as you stir), and check to see that the raisins are evenly distributed.
Step 4Pour the mixture into a lightly buttered 13-by-9-inch baking dish. Lightly scatter and press the pecans onto the top of the pudding.
Step 5Bake the pudding until it has puffed and the custard is set in the center, about 70 to 90 minutes. Remove the pudding to a rack to cool for 2 hours before serving.
Step 1In a medium saucepan, heat the corn syrup, rum and vanilla to a simmer over medium heat. Remove from heat and, using a whisk, vigorously beat in the softened butter. Keep at room temperature until ready to serve. This makes about 3 cups sauce.