Preheat your oven to 170°C (338°F). Grease or line a 20x20 cm (8x8 inch) baking pan.
In a mixing bowl, beat the butter and sugar together until light and creamy.
Add the vegetable oil and mix well. Then add the eggs one at a time, mixing after each addition. Stir in the vanilla extract and plain yogurt.
Sift in the plain flour and baking powder. Gently fold everything together until the batter is smooth. Do not overmix.
Pour the batter into the prepared pan and level the top. Bake for 27 minutes, or until a toothpick inserted in the center comes out clean.
Remove from the oven and let the cake cool completely before adding the icing.
In a small bowl, mix the powdered sugar with 1 to 2 tablespoons milk until smooth. Adjust the milk slowly until you get a thick but pourable consistency.
Drizzle the icing over the cooled cake. Let it set slightly, then slice and serve.
