2 sticks (1 cup) unsalted butter, softened, plus additional for greasing pan
1 cup self-rising cake flour plus additional for flouring pan
1 cup pitted dates (5 oz), finely chopped
1 1/4 cups packed dark brown sugar
1 large egg
Put oven rack in middle position and preheat oven to 350°F. Butter and flour an 8- by 2-inch round cake pan.
Simmer dates in 1 cup water in a 1-quart heavy saucepan, covered, until soft, about 5 minutes. Let stand, covered, off heat 5 minutes.
Beat together 1 stick butter and 1/4 cup brown sugar in a large bowl with an electric mixer at medium-high speed until light and fluffy, about 4 minutes. Beat in egg until combined. Add flour and 1/8 teaspoon salt and mix at low speed until just combined. Add dates and mix until just combined well.
Pour batter into pan and bake until a wooden pick or skewer inserted in center comes out clean, about 30 minutes.
Meanwhile, melt remaining stick butter in a 2-quart heavy saucepan over moderate heat and stir in remaining cup brown sugar, 1/3 cup water, and a pinch of salt. Boil over moderately high heat, uncovered, stirring occasionally, until sugar is dissolved and sauce is reduced to about 1 1/4 cups, 2 to 8 minutes. Remove from heat and cover.
Transfer pudding in pan to a rack and poke all over at 1-inch intervals with a chopstick. Gradually pour half of warm sauce evenly over hot pudding. Let stand until almost all of sauce is absorbed, about 20 minutes.
Run a thin knife around edge of pan. Invert a plate over pudding and invert pudding onto plate. Pour remaining warm sauce over pudding and serve immediately.