Preheat oven to 350°. Brush a 9"-diameter cake pan with extra-virgin olive oil. Line bottom with a parchment paper round. Combine 1⅓ cups blackberries, raspberries, blueberries, or halved or quartered hulled strawberries and 2 Tbsp. (25 g) sugar in a medium bowl. Using a fork, mash together until sugar is hydrated and no dry spots remain. There will be some totally smashed berries, but some should still be intact. Set berry mixture aside.

Whisk 1½ cups (188 g) all-purpose flour, 1½ tsp. baking powder, 1 tsp. Diamond Crystal or ½ tsp. Morton kosher salt, and ½ tsp. baking soda in a medium bowl to combine. Set aside.

Place 1 Tbsp. (heaping) finely grated lemon zest and remaining 1 cup (200 g) sugar in a large bowl. Using your fingers, work lemon zest into sugar until sugar starts to clump and mixture is very fragrant. Add 3 large eggs, 2 tsp. vanilla extract, and remaining ½ cup extra-virgin olive oil and whisk vigorously until mixture is pale and smooth. Add juice of 1 large lemon and 8 oz. whole-milk fresh ricotta and mix just until combined. Add reserved dry ingredients and mix with a rubber spatula until incorporated and batter is mostly smooth.

Scrape batter into prepared pan and spoon reserved (now syrupy) berries on top. Bake cake until golden brown on top and a tester inserted into the center comes out clean, 45–50 minutes. Transfer to a wire rack; let cool in pan. Invert cake onto a large plate; peel away parchment and discard. Turn cake right side up. Do ahead: Cake can be baked 2 days ahead. Store tightly wrapped at room temperature.