4 tablespoons (1/2 stick) unsalted butter
1 cup unbleached all-purpose flour
3/4 cup packed light brown sugar
2 teaspoons baking powder
1/2 teaspoon table salt
1/2 cup whole milk
2 teaspoons pure vanilla extract
Sweetened whipped cream or vanilla or coffee (my favorite) ice cream (optional)
Place an oven rack in the middle of the oven and preheat the oven to 350ºF. Lightly grease an 8-cup shallow baking dish or pan. Melt the butter in the microwave or in a small saucepan over low heat. Bring 1 1/4 cups water just to a boil over high heat.
Stir together the flour, brown sugar, baking powder, and salt in a medium bowl until there are no brown lumps. Add the milk, the melted butter, and vanilla. Stir just until combined; transfer the stiff batter to the greased baking dish.
Set the baking dish in the oven and carefully pour the boiling water over the surface of the batter. Bake for 30 to 35 minutes, or until the cake on the top has a crisp, golden surface and the pudding sauce on the bottom bubbles (sometimes through the top surface).
Set aside for 5 to 10 minutes to cool slightly. Serve warm with whipped cream or ice cream, if desired.