Heat the oven to 350 F / 175 C with a rack in the lower third. Fit the crust into a 9-inch / 23 cm pie plate, crimp the edge, and set the plate on a rimmed 33 x 46 cm / 13 x 18 inch sheet pan.
Whisk the sugar, cocoa, salt, and water in a 2-quart / 1.9 L saucepan. Heat over medium, stirring, for 2 to 3 minutes until smooth and just bubbling at the edges. Stir in the vanilla, cool for 2 to 3 minutes, then pour the sauce into the unbaked crust.
Whisk the flour, sugar, baking powder, and salt in a large bowl. Add the softened butter, milk, egg, and vanilla. Whisk for about 1 minute until the batter is smooth and thick.
Drop the batter over the sauce in spoonfuls, working from the edge toward the center. Smooth it gently with the back of the spoon. Don't stir or pour hard, and a few chocolate gaps are fine.
Bake for 40 to 45 minutes until the top is golden, springs back lightly, and a toothpick poked just into the cake layer comes out without wet batter.
Cool the pie on a wire rack for 60 minutes so the sauce layer sets. Slice into 8 wedges and serve warm or at room temperature.