Preheat the oven to 350°F. Grease a 9×5-inch or 8 1/2 x 4 1/2-inch loaf pan lightly with olive oil and line it with parchment paper, leaving about a 1-inch overhang on the long sides, to make a sling.
Crack the eggs into a large bowl, add the sugar, and whisk vigorously until pale yellow and frothy, about 1 minute. Whisk in the yogurt, oil, and vanilla extract.
To the bowl with the wet ingredients, whisk in the baking powder and kosher salt. Add the flour and stir with a rubber spatula until just combined and all of the flour is incorporated. (A few small lumps in the batter are fine, be sure to not overmix.)
Transfer the batter to the prepared pan and use a spatula to spread it out evenly. Bake until the cake is lightly browned and a cake tester inserted into the center comes out clean with just a few crumbs, about 45 minutes.
Transfer the pan to a cooling rack and let cool for 10 minutes before using the parchment paper sling to lift the cake out of the pan and onto the cooling rack to cool completely.
Dust with powdered sugar, if desired, before slicing and serving and enjoy!