Make the biscuits:
Line a large rimmed baking sheet with parchment.
Chill the butter pieces in the freezer for about 15 minutes. Meanwhile, in a large mixing bowl, whisk the flour, sugar, baking powder, cinnamon, salt, and nutmeg. With a pastry blender, cut in the cold butter until the mixture resembles a very coarse meal strewn with pieces of butter the size of peas and pistachios. Add the walnuts and toss with a fork to distribute them.
Combine the cream with the vanilla in a liquid measuring cup. Make a well in the center of the flour mixture and pour the cream into the well. With a fork, work your way around the bowl, pulling the dry ingredients into the wet and mixing until the ingredients are mostly moistened and a rough dough forms. Gently knead the dough a few times in the bowl to pick up any dry ingredients in the bottom of the bowl and bring it together into a loose ball. Transfer the dough to the parchment-lined baking sheet and pat into a 3/4-inch-thick circle. Refrigerate for about 20 minutes.
Position a rack in the center of the oven and heat the oven to 450°F.
Dip a 3-inch round plain-edge biscuit cutter in flour and cut straight down through the dough without twisting to form the biscuits. (Dip the cutter in flour before cutting each biscuit.) Carefully move the biscuits to one end of the baking sheet. Gather the dough scraps together, knead briefly to smooth the dough, and pat into a 3/4-inch-thick circle on the other end of the baking sheet. Cut out more biscuits. Repeat one more time. You can use the remaining scraps to make an additional biscuit or two, but those won't have as nice a texture as the others.
Spread the biscuits out over the baking sheet. Lightly brush their tops with cream and then sprinkle with the demerara sugar.
Put the baking sheet in the oven and lower the temperature to 425°F. Bake until the biscuits are deeply browned on top and bottom, 20 to 25 minutes. Let them cool on a rack while you make the caramelized plums.
Make the whipped cream:
Using a chilled bowl and beaters, whip the heavy cream with a hand-held or stand mixer on medium speed until it thickens slightly. Add the sour cream and confectioners’ sugar and whip on medium-high speed until it holds medium peaks. Use right away.
Assemble the shortcakes:
Using a fork, split each biscuit in half horizontally. Set the bottom halves on serving plates and spoon 5 or 6 plum wedges onto each biscuit. Dollop the cream on the plums and drizzle with some of the caramelized plum juice. Perch the tops of the biscuits on the cream. If you have plums left, arrange them on the plates alongside the shortcakes.