Adjust an oven rack to the lower third position and preheat the oven to 400 degrees. Butter a 13 x 9 x 2-inch baking pan and set aside. Do not use cooking spray because the dough must adhere to the pan when you spread it.
In a medium-size bowl, whisk together thoroughly the flour, salt, baking powder, nutmeg, and 1/3 cup sugar. Cut the 5 tablespoons butter into the dry ingredients with a pastry blender until the fat is in smallish pieces and looks like coarse crumbs.
In a small bowl beat the egg with a table fork well to combine the white with the yolk. Stir in the milk and vanilla, and pour into the dry ingredients. Stir with the fork until the dough is thoroughly moistened. It will be very thick.
To spread the dough into a thin even layer in the pan, place small spoonfuls of it into the buttered pan, spacing them an inch or two apart (pretend you're making drop cookies), forming about a dozen or so little mounds. Spread them together with the back of a teaspoon to make a very thin even layer covering the bottom of the pan.
Take your time and have patience with this step. Swirling the back of the spoon back and forth in a figure eight pattern as you spread the batter does the best job.
Arrange the plums, cut sides down, in tightly packed rows on top of the batter. (I usually get 5 or 6 half-plums across the short side of the pan and 10 to 12 along the long side). Brush the plums with the melted butter. Combine the 1/4 cup sugar with the cinnamon and sprinkle evenly over the plums.
Bake for 35 to 40minutes, until the fruit juices have almost stopped bubbling and a toothpick stuck into the cake portion comes out clean. Cool the kuchen in its pan on a wire rack. Cut around the sides of the kuchen to release from the pan. Serve portions warm or at room temperature.
NOTE: If you want to freeze the kuchen, butter the baking pan and line it with aluminum foil. Butter the foil and proceed with the recipe. After the kuchen has cooled in its pan, grasp the edges of foil and carefully lift the cake out of the pan. Set the cake on a cookie sheet and place it in the freezer. When solidly frozen, peel off the foil, wrap the cake tightly in plastic wrap, and enclose the cake securely in heavy-duty foil. Date the kuchen and store it in the freezer for up to 6 months. To reheat, unwrap the frozen cake (remove both the foil and plastic wrap), and set the kuchen on a baking sheet. Tent it loosely with the foil. Place the frozen cake into a preheated 300 degree oven for 20 to 30 minutes, until it is completely thawed and slightly warm. Cut into portions and serve.