Preheat the oven to 375 degrees F. Combine 3/4 cup sugar and 1 1/2 tablespoons water in a 10-inch ovenproof skillet and stir until the mixture looks like wet sand. Cook over medium-high heat, gently swirling the pan occasionally but not stirring, until light amber, about 5 minutes. Remove from the heat and arrange the plums in concentric circles over the caramel; set the skillet aside.
Whisk the flour, baking powder, baking soda and salt in a bowl. Beat the butter and the remaining 1 cup sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 3 minutes. Add the eggs one at a time, beating after each addition, then beat in the vanilla. Whisk the milk, sour cream and lemon zest in a small bowl. Reduce the mixer speed to low; add the flour mixture to the butter mixture in 3 batches, alternating with the milk mixture, beginning and ending with the flour, and mix until just incorporated. Pour the batter over the plums and spread evenly.
Transfer the skillet to the oven and bake until the cake is golden and a toothpick inserted into the center comes out clean, about 50 minutes. (Cover loosely with foil if the cake is browning too quickly.) Let sit 15 minutes, then invert onto a platter and let cool completely.