Preheat oven to 350°F (175°C). Grease an 8×8-inch (20×20 cm) baking pan and line with parchment paper for easy removal.

In a bowl, whisk together flour, cocoa powder, baking powder, baking soda, and salt until evenly combined.

In a larger bowl, beat the softened butter and granulated sugar until light and fluffy, about 2–3 minutes. Add eggs one at a time, beating well after each addition. Stir in vanilla.

Alternate adding the dry mixture and warm milk to the butter mixture, beginning and ending with the dry ingredients. Fold in sour cream and then gently fold in chopped dark chocolate. Batter should be thick but pourable.

Pour batter into the prepared pan, smooth the top, and bake for 25–30 minutes, or until a toothpick inserted into the center comes out with a few moist crumbs (not wet batter). The top will feel set and spring back slightly.

While the pudding bakes, combine brown sugar and butter in a small saucepan over medium heat. Stir until the sugar dissolves and the mixture is bubbling. Carefully stir in heavy cream and simmer for 2–3 minutes until the sauce thickens slightly. Remove from heat and stir in vanilla and a pinch of salt. Sauce should be glossy and pourable.

When the pudding is out of the oven, poke holes across the surface with the handle of a wooden spoon. Pour about two-thirds of the warm toffee sauce evenly over the hot pudding so it soaks in. Reserve the rest of the sauce for serving.

Let the pudding rest for 5 minutes so the sauce can sink into the crumb. Cut into squares, spoon extra sauce over each portion, and add a scoop of vanilla ice cream or dollop of whipped cream if desired.