Place chopped chocolate and butter in a large heatproof bowl. Set aside.
Sift cornstarch, sugar, cocoa powder and salt into a heavy saucepan. Slowly whisk in milk and stir until smooth. Place over medium heat and cook for about 15 minutes, stirring occasionally, until the mixture is close to a simmer and is thick enough that it clings to the whisk. (Similar to thick batter. As it thickens, you’ll feel it get heavier as you stir.)
Remove from heat and immediately pour pudding over the chocolate. Add the vanilla and keep stirring until the chocolate and butter are completely melted and the pudding is smooth.
Pour some of the pudding into prepared pudding bowls. Add about 2-3 chopped ladyfingers to each one. Top with the rest of the pudding. If any cookies are peaking out, dunk them back in with a spoon.
Chill pudding until it’s set and cool to the touch, at least 3 hours. Cover bowls with plastic wrap and chill in the fridge overnight, if need be.