Preheat the oven: Set oven to 350°F (175°C). Grease and line an 8-inch (20 cm) round cake pan with parchment paper.
Mix dry ingredients: In a medium bowl, whisk together flour, cocoa powder, baking powder, and salt until well combined.
Combine wet ingredients: In a separate large mixing bowl, whisk oil, sugar, and yogurt until smooth. Stir in vanilla extract.
Add milk gradually: Alternate adding the dry ingredients and milk to the wet mixture, stirring until just combined. Be careful not to overmix—the batter should be smooth but not overworked.
Bake the cake: Pour batter into prepared cake pan and bake for 30–35 minutes, or until a toothpick inserted in the center comes out clean. Let cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.
Prepare the frosting: In a small saucepan, heat milk until just simmering. Remove from heat and add chopped chocolate. Stir until smooth and glossy. Let cool slightly to thicken.
Frost the cake: Once the cake is fully cooled, spread the frosting evenly over the top and sides. Slice and serve.