Maybe cake is not the right term for this dessert-we don't serve it in slices, but scoop it still warm into bowls and add a topping of ice cream or sweetened whipped cream. Italian prune plums come into markets in late August and stay through September. They are perfect for late-summer baking, since they get jammy but still hold their shape when cooked. If you have vanilla or almond extract on hand, stir 1/4 teaspoon of it into the egg mixture.
Preheat the oven to 350 degrees F. Generously butter the bottom and sides of a 9 1/2-inch deep-dish pie pan. Use a fork to beat the egg with 1/4 cup water in a small bowl until combined.
Put the flour and sugar in the bowl of a food processor and pulse until the mixture is uniform. Drop in the butter cubes, toss with a fork to coat in the flour mixture and pulse until the mixture is crumbly and moist, about 8 long pulses. Add half of the egg mixture and process about 15 seconds. Add the remaining egg mixture and process an additional 15 seconds. The batter should be homogenous but not too thick. (Don't worry if it curdles a bit because of the brown sugar. It won't affect the batter.)
Spread the batter in the prepared pan. Very lightly press the plums into the batter. Bake until the plums are tender and a cake tester or toothpick inserted in the middle of the cake comes out clean, about 30 minutes. Cool slightly and serve with whipped cream or ice cream if desired.
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