Preheat the oven to 170°C (340°F).
Lightly grease a square baking pan (approximately 22cm or 8.5 inches).
Mix together soy milk and vinegar; set aside to thicken (this substitutes for dairy buttermilk).
Mix in the remaining wet ingredients: yogurt and olive oil.
Sift in the dry ingredients: flour, cocoa powder, baking powder, sugar, and salt.
Gently mix until just combined.
Pour the cake mixture into the greased baking pan.
Sprinkle the light brown sugar and cocoa powder over the cake batter.
Carefully pour the boiling water over the back of a spoon onto the pudding (this prevents the batter from splitting).
Bake for 30-35 minutes, or until the center is almost set.
Allow to cool slightly before serving with ice cream and an extra drizzle of olive oil.