Preheat your oven to 350°F. Grease a 9-inch round cake pan with extra-virgin olive oil and line the bottom with a round of parchment paper.
In a medium bowl, mix together 1⅓ cups of your choice of berries with 2 tablespoons of sugar. Use a fork to mash the berries until the sugar is dissolved, leaving some berries intact. Set this mixture aside.
In another medium bowl, whisk together 1½ cups of all-purpose flour, 1½ teaspoons of baking powder, 1 teaspoon of kosher salt, and ½ teaspoon of baking soda. Set this dry mixture aside as well.
In a large bowl, combine 1 tablespoon of finely grated lemon zest with the remaining 1 cup of sugar. Use your fingers to rub the lemon zest into the sugar until it becomes fragrant and starts to clump.
Add 3 large eggs, 2 teaspoons of vanilla extract, and ½ cup of extra-virgin olive oil to the sugar mixture. Whisk everything together vigorously until the mixture is pale and smooth.
Stir in the juice of 1 large lemon and 8 ounces of whole-milk fresh ricotta until just combined.
Gently fold in the reserved dry ingredients using a rubber spatula until the batter is mostly smooth.
Pour the batter into the prepared cake pan and spoon the reserved berry mixture on top.
Bake the cake for 45 to 50 minutes, or until the top is golden brown and a tester inserted into the center comes out clean.
Once baked, transfer the cake to a wire rack and let it cool in the pan. After cooling, invert the cake onto a large plate, remove the parchment paper, and turn the cake right side up.
If desired, you can bake the cake up to 2 days in advance. Just store it tightly wrapped at room temperature.
