Preheat the oven to 350 degrees F. Grease a 9-inch springform pan with 1 tablespoon of the butter and dust it with 1 tablespoon of the flour.
Toss the plums with 2 tablespoons of the sugar and the brandy; set aside.
Cream the remaining 8 tablespoons butter in a large bowl with 1 cup of the sugar, lemon zest, and vanilla until light and fluffy. Sift the remaining 1 cup flour, baking powder, and salt together and beat into the creamed butter.
In a separate bowl, beat the eggs until they start to foam. Do not over beat, or the cake will be tough. Add the eggs and ground walnuts to the flour and butter mixture. Mix well.
Scrape the batter into the prepared pan. Arrange the plums on top in rings. Sprinkle with any remaining brandy-sugar syrup and the remaining 2 tablespoons sugar.
Bake for 1 hour, or until the cake is golden brown on top and a toothpick inserted into the cake (not the plums) comes out clean. Let cool for 10 minutes before removing the sides of the springform pan and serving. Garnish with a dusting of confectioners' sugar.