Preheat the oven to 350 degrees Fahrenheit.
Line a 9-inch square cake pan with parchment paper.
In a medium bowl whisk together flour and baking powder. Set aside.
In the bowl of a stand mixer, combine the eggs and sugar. Beat on medium-high speed until the mixture is pale, thick, and has doubled in volume — this should take about 7 minutes.
Add yogurt, cooled melted butter (or oil) and vanilla and beat on low speed until combined.
Add the flour in two additions, mixing on low speed just until incorporated. Be careful not to overmix — you want to keep the batter light and airy.
Transfer the batter into the prepared cake pan. Smooth out the top.
Gently arrange the halved strawberries on top of the batter, cut side down, pressing them in slightly to evenly cover the surface of the cake.
Bake for 45 mins at 350 degrees or until a toothpick inserted in the center comes out clean.
Cool on a wire rack for at least 20 minutes, then remove from the pan holding by the parchment paper.
Sprinkle with powdered sugar, if you want. Serve still warm or at room temperature.