Preheat the oven to 350 degrees . Grease a 5-by-9-inch loaf pan and set aside. Using a standing electric mixer, beat 1/2 cup sugar and the butter at high speed until creamy, about 3 minutes. Add the pumpkin pie spice, ginger, baking powder, baking soda and salt and mix at low speed until blended. Beat in the egg at medium speed. Add the sour cream, molasses and vanilla and mix at low speed until incorporated. Add the flour and mix at low speed just until blended, then drizzle in 1/4 cup hot water and mix just until smooth.
Pour the batter into the prepared pan and bake until the gingerbread cake is springy and a toothpick inserted in the center comes out clean, 45 to 50 minutes. Transfer to a rack to cool for 10 minutes, then remove from the pan and let cool completely.
Line a 5-by-9-inch loaf pan with 2 overlapping sheets of plastic wrap, allowing a 4-inch overhang on all sides; freeze the pan.
Using a food processor, pulse 2 cups plums and the vanilla ice cream until the fruit is chopped and mixed in. transfer half the mixture to a bowl and place in the freezer. Remove the loaf pan from the freezer and spread the remaining mixture in an even layer on the bottom.
Trim the short ends of the gingerbread and discard. Cut the cake crosswise into 1/2-inch slices. Add slices of the cake to fit evenly in a single layer in the pan, trimming as necessary. Press the cake down gently onto the ice cream. Freeze for 30 minutes.
Spoon the reserved ice cream-plum mixture on top of the cake layer. Cover with a second layer of cake, then wrap with the overhanging plastic. (Keep any leftover cake for snacking.) Freeze for at least 4 hours.
Meanwhile, in a medium skillet, combine the remaining plums and 1/3 cup sugar and cook, stirring often, over medium-low heat, until tender, about 10 minutes. Let cool for 10 minutes, then stir in the lemon juice.
Unwrap the top of the cake and invert the cake onto a serving platter. Peel off the plastic wrap. Drizzle half the plum sauce over the loaf. Cut and serve with the remaining plum sauce on the side.