For the cake:
1/2 pound dates, pitted and coarsely chopped
7 tablespoons unsalted butter, softened, plus extra for the baking dish
1 cup sugar
1&3/4 cups all-purpose flour
1 teaspoon baking powder
1 teaspoon baking soda
1 teaspoon vanilla extract
1 egg, lightly beaten
For the topping:
3/4 cup light brown sugar
6 tablespoons unsalted butter
5 tablespoons heavy cream
Make the cake:
1. Place the dates in a heatproof bowl. Bring 1 cup water to a boil in a small saucepan, pour it over the dates, and let them soak until plumped, 10 minutes. Do not drain.
2. Preheat the oven to 375°F. Coat a 7-by-11-inch baking dish with butter.
3. Cream the butter and sugar together in a mixer. Sift the flour and baking powder together in a bowl and set aside. Combine the dates and their soaking liquid, the baking soda, and the vanilla in a small bowl.
4. With the mixer on medium speed, add the egg and one third of the flour mixture to the creamed butter and beat until incorporated, about 2 minutes. Gradually add the remaining flour and beat until it is mixed into the batter. Add the date mixture and beat until well blended.
5. Pour the batter into the prepared baking dish and bake until the cake springs back when tapped with a finger, or when a cake tester inserted in the middle comes out clean, about 30 minutes. Using a fork, poke holes all over the surface of the cake.
6. Preheat the broiler.
Make the topping:
1. Combine the butter, brown sugar, and cream in a saucepan and cook over medium heat, stirring, until it bubbles slightly, about 3 minutes.
2. Pour this caramel syrup over the baked cake and then slide it under the broiler until the surface begins to bubble, about 30 seconds. Immediately remove the dish from the oven. Let the cake cool a bit; then cut it into squares. Serve warm or at room temperature.