To make the cake: Preheat oven to 350 degrees F. Grease an 8-inch springform pan with butter.
In a stand mixer fitted with a paddle attachment, cream together the butter and brown sugar, until light and fluffy, about 3 minutes. Add the eggs, one at a time, scraping down the bowl after each. Mix in the zest and vanilla. In a separate bowl whisk together the flour, baking powder and salt. Add the dry ingredients to the butter/sugar mixture and mix on low speed until just combined. Spread half of the batter into the prepared pan.
Place half of the cut plums on the batter, cut-side up. Spread the remaining batter over the plums. Spread the batter into an even layer and top with remaining plums.
To make the topping: In a bowl, mix together the flour, sugar, salt, cinnamon and melted butter. You may have to mix it with your hands to get all the butter incorporated.
Spread the crumb topping over the plums evenly.
Place the cake on two stacked cookie sheets, to prevent the bottom from getting too brown during the long baking time. Bake for about 1 hour 15 minutes or until a tester come out clean. Cool on a rack.
Serve it slightly warm or at room temperature.