6 tablespoons (3/4 stick) unsalted butter, at room temperature, plus extra for greasing the dish
10 to 12 purple "prune" plums, cut in half and pitted (I used whatever the plums from our tree are.
1 3/4 cups granulated sugar, divided
2 large eggs, at room temperature
1/3 cup sour cream
1/2 teaspoon grated lemon zest
1/2 teaspoon pure vanilla extract
1 cup plus 2 tablespoons all-purpose flour
1/2 teaspoon baking powder
1/4 teaspoon kosher salt
Preheat the oven to 350 degrees F. Generously butter a 9-inch glass pie dish and arrange the plums in the dish, cut side down.
Combine 1 cup of the granulated sugar and 1/3 cup water in a small saucepan and cook over high heat until it turns a warm amber color, about 360 degrees F on a candy thermometer. Swirl the pan but don't stir. Pour evenly over the plums.
Meanwhile, cream the 6 tablespoons of butter and the remaining 3/4 cup of granulated sugar in the bowl of an electric mixer fitted with the paddle attachment, until light and fluffy. Lower the speed and beat in the eggs one at a time. Add the sour cream, zest, and vanilla and mix until combined. Sift together the flour, baking powder, and salt and, with the mixer on low speed, add it to the butter mixture. Mix only until combined.
Pour the cake batter evenly over the plums and bake for 30 to 40 minutes, until a cake tester comes out clean. Cool for 15 minutes, then invert the cake onto a flat plate. If a plum sticks, ease it out and replace it in the design on top of the cake. Serve warm or at room temperature, dusted with confectioners' sugar.