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Preheat the oven to 350°F with a rack in the center. Grease an 8" square pan with nonstick spray.
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To make the batter: In a large bowl, whisk the butter, brown sugar, and vanilla together until the sugar has dissolved, less than 1 minute.
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Add the egg and whisk until completely incorporated. Whisk in the sourdough discard and buttermilk until no traces remain.
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Using a flexible spatula, stir the flour, cocoa, baking powder, salt, and baking soda into the wet ingredients. Transfer the batter to the prepared pan and use a small offset spatula to spread it evenly.
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To make the chocolate sauce: In a medium microwave-safe bowl or liquid measuring cup with at least a 2-cup capacity, place the chopped chocolate. Heat in the microwave in 30-second increments, stirring between each, until the chocolate is melted, 1 1/2 to 2 minutes.
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Whisk the water, brown sugar, and espresso powder into the melted chocolate; the sauce will be very thin and fluid.
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To assemble the chocolate pudding cake: Pour the chocolate sauce on top of the batter.
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If using a metal pan, bake the cake for 24 to 28 minutes; if using a glass or ceramic pan, bake for 30 to 35 minutes, until the middle has just set. A toothpick or thin knife inserted into the center of the cake should have a few wet crumbs on it.
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Remove the cake from the oven and let it cool in the pan on a wire rack for 5 minutes.
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Serve the chocolate pudding cake while it’s still warm, with a dollop of fresh whipped cream or ice cream on top of each portion, if desired.
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Storage information: Let the cake cool completely then cover the pan with plastic wrap, or transfer leftovers to an airtight container and refrigerate for up to 3 days. (As it rests, the cake will absorb the chocolate sauce and become more fudgy instead of saucy.) Reheat slices in a microwave-safe bowl for 10 to 15 seconds.
