Make the ice cream: To a large bowl, add the crumbled Christmas pudding and set aside. In a medium bowl, whisk together the egg yolks and sugar until thick and pale. In a medium saucepan, bring the milk and cream to a boil over medium-high heat, then whisk a little of the milk and cream into the yolk mixture. Once the bottom of the bowl is warm, whisk in the remaining milk and cream, then return the milk-egg mixture to the saucepan and set it over low heat. Cook, using a wooden spoon to stir constantly, until the custard is thick enough to coat the back of a spoon and reaches 180 degrees F on an instant-read thermometer. Turn off the heat and add the glycerin, if using, to help prevent ice crystals forming during freezing. Pour the mixture over the Christmas pudding and stir to combine.
Set the ice cream base aside to cool to room temperature, then cover the bowl with plastic wrap and refrigerate until thoroughly chilled (at least 4 hours). Pour the mixture into an ice cream maker and churn and freeze the ice cream according to manufacturer's instructions.
Make the cake: To a large saucepan add the unpeeled oranges, cover with cold water, and bring the liquid to a boil over high heat. Reduce the heat to medium-low, cover and simmer for until the orange rinds are soft and the insides have broken down, 1 1/2 to 2 hours. Drain the water and, when cool enough to handle, cut each orange in half and discard the seeds. Add the oranges to a food processor (skins, pith and fruit) and pulse a few times until rough textured and well chopped.
Preheat the oven to 325 degrees F and grease and line a 9 1/2-inch loaf pan or an 8-inch cake pan. Weigh the orange pulp and keep 1 pound, discard the rest. Return the pulp to the food processor with the almond meal, superfine sugar, baking powder, ginger and cinnamon. With the food processor running, add the eggs one at a time until the batter is smooth.
Pour the cake mixture into the prepared pan and bake until the cake sides pull away from the pan, 1 hour to 1 hour and 20 minutes (check the cake after about 40 minutes and if the top of the cake is becoming too dark, loosely cover it with aluminum foil). Remove the cake from the oven and set on a wire rack to cool. Once the cake has cooled completely, invert the cake onto a cake plate.
Make the sauce: To a medium saucepan, add the plums, butter, sugar, cinnamon and lemon juice and set over low heat until everything comes to a simmer. Continue to cook until the plums are very soft and almost turn into a puree. Remove and discard the cinnamon stick and transfer the plum mixture to a food processor or blender and puree until smooth. Pass the sauce through a fine-mesh sieve and into a medium bowl and set aside until needed. (If the puree is a little thick, add some water.)
Serve the cake in slices with a scoop of ice cream and a drizzle of the sauce.
Glycerin is a sweet syrup that can be added to ice cream to help keep it soft and creamy. It can be purchased in baking supply stores or online.