Combine the blackberries, raspberries, lemon juice, and sugar in a medium bowl. Stir gently to lightly coat the berries without mashing them. Cover and place in the fridge.: The berries should smell bright and fresh as you mix in the lemon juice and sugar , and you will notice a faint glossy sheen as the sugar begins to draw out juices. This short rest in the fridge concentrates flavors and prevents the fruit from bleeding into the batter when served, which preserves pretty presentation. One common mistake is stirring too vigorously and mashing the fruit, which yields a muddy topping. Keep your motions gentle and check the bowl visually for intact berries and a slight syrup forming at the bottom.
Preheat the oven to 350°F (175°C) and grease a 9-inch (23 cm) springform pan.: As the oven comes to temperature you should smell the dry, warm air from the preheating chamber, an olfactory cue that it is ready. Proper greasing prevents sticking and ensures clean removal, so coat the pan evenly with oil or butter and add a parchment round if you like. A mistake to avoid is skipping the greasing step, which can tear the cake when unmolding; also ensure the oven actually reaches temperature before baking, because underheated ovens slow rise and alter texture.
In a medium mixing bowl, whisk together the yogurt, eggs, sugar, and vanilla. Add the flour, baking powder, and salt, and stir just until combined. Using a spatula, fold in the oil until you have a thick, shiny batter.: As you stir, notice the batter becoming glossy and cohesive, with no streaks of flour. The initial whisking dissolves the sugar slightly and blends the eggs for lift, while the gentle folding of extra virgin olive oil preserves air bubbles that help the cake stay airy. Overmixing after adding the flour is a common error that can develop gluten and toughen the crumb; stop mixing as soon as the components are homogenous, and the batter should still be slightly fluid yet weighty on the spatula.
Pour the batter into the prepared pan and bake for 35–40 minutes, or until a toothpick inserted into the center of the cake comes out clean. Allow the cake to cool for 10 minutes before removing it from the pan.: While baking you will see the top become pale golden and the edges pull slightly away from the pan, and you may notice a gentle dome forming. The fragrance will shift from a raw batter scent to warm, baked aroma that hints at vanilla and olive oil. Use the toothpick check near the center to confirm doneness; if the toothpick shows wet batter, return the cake to the oven for a few more minutes. A frequent mistake is opening the oven repeatedly to check, which lets heat escape and can cause uneven rise; try to minimize door openings and rely on visual cues.
To serve the cake, lightly dust the top with powdered sugar and mound the berries on top.: The powdered sugar will add a soft matte contrast to the glossy berries, and piling the fruit creates an inviting burst of color and texture. As you arrange the berries, you can spoon any accumulated syrup over the slices for extra shine. Beware of adding the berries too early while the cake is still warm, because the heat will soften them excessively and make the topping runny; a brief cooling period preserves structure and presentation.
Cut the cake into slices and serve with whipped cream, if desired.: When slicing, use a sharp knife and make clean strokes to avoid compressing the crumb; the cake should release with a tender pull and display a softly set interior. If you choose to add whipped cream, it adds a creamy counterpoint to the fruit and enhances the sense of indulgence. A common misstep is cutting the cake while it is still piping hot, which can cause collapsing or crumbly slices, so wait until it is warm rather than hot for the best texture.