Make the streusel:
Put the flour, sugar, kosher salt, and cardamom in a small mixing bowl and stir the ingredients with a fork until thoroughly combined. Drizzle the melted butter over the mixture and stir with the fork until the mixture resembles a clumpy dough. Using your fingers, break the mixture into pistachio-size clumps and large crumbs. (If the streusel is sandy and won’t clump, add a little more melted butter, 1 tsp. at a time.) Refrigerate the streusel while you prepare the cake batter.
Make the cake:
Position a rack in the center of the oven and heat the oven to 375°F. Lightly butter and flour an 8x8x2-inch straight-sided cake pan.
Beat the eggs lightly in a small mixing bowl. Whisk in the granulated sugar, brown sugar, milk, and vanilla until well blended. Set aside.
In a large mixing bowl, whisk the flour, baking powder, table salt, and cardamom until well blended. Add the butter pieces to the bowl and cut them into the flour with a pastry blender until the mixture resembles a very coarse meal strewn with pieces of butter the size of small peas and oat flakes.
Add the egg mixture to the flour mixture. With a wooden spoon, fold and stir the ingredients together until it’s a thick batter speckled with visible lumps of butter, 45 seconds to 1 minute.
Scrape the batter into the prepared pan and spread it evenly. Break up the streusel mixture with your fingers and sprinkle half of it evenly over the batter. Arrange the plum quarters skin side down on the batter, with each piece at a 45-degree angle to the sides of the pan. Sprinkle the remaining streusel evenly over the cake.
Bake the cake for 20 minutes and then rotate the pan. Continue baking until the top of the cake is golden brown and a toothpick inserted in the center comes out with a few moist crumbs clinging to it, 15 to 20 minutes more.
Cool the cake in its pan on a rack for at least an hour before cutting. Serve warm or at room temperature.