Preheat the oven to 350°F (175°C). Grease a 9-inch round cake pan and line the bottom with parchment paper.
Zest the lemons, then juice them; set the zest and juice aside separately.
In a large bowl, whisk together the olive oil, sugar, eggs, lemon zest and lemon juice until smooth and slightly thickened.
Whisk in the yogurt and vanilla extract until combined.
In a separate bowl, whisk together the flour, baking powder, baking soda and salt.
Add the dry ingredients to the wet ingredients and mix just until no streaks of flour remain; do not overmix.
Pour the batter into the prepared pan and smooth the top.
Bake for 45-50 minutes, until the top is golden, the edges pull slightly away from the pan, and a tester inserted in the center comes out with a few moist crumbs.
Cool in the pan for 15 minutes, then turn out onto a rack to cool completely, about 20-25 minutes more, before dusting with powdered sugar and slicing.