Preheat oven to 350℉.
Generously grease a loaf pan with butter and set aside.
Sift together flour, baking powder, and salt into a bowl. Set aside.
Zest the lemons and divide their juice into two bowls.
Combine the sugar and the lemon zest in a bowl. Rub the lemon zest into the sugar with your fingers until it's fragrant, and the mixture resembles coarse sand.
Beat together the eggs, yogurt, juice of 1 lemon, and the vanilla extract until well blended.
Add the flour mixture and then the sugar mixture; beat on low speed until batter is combined.
Fold in olive oil using a spatula, until combined. Do not over mix.
Pour the batter into the prepared loaf pan.
Bake in the preheated oven until a toothpick inserted into the center of the loaf comes out clean, about 1 hour to 1 hour and 15 minutes.
Remove from the oven and allow loaf to cool for 30 minutes in the loaf pan. After 30 minutes, run a knife around the edge of the pan to loosen it and transfer it to a serving dish. Allow to cool completely.
Once loaf has cooled completely, whisk the remaining lemon juice and confectioners sugar together in a bowl to make a thin glaze. Drizzle the glaze over cake.
Allow glaze to set before slicing into the loaf.