Preheat oven to 350 degrees F (175 degrees C). Grease an 8-1/2x4-1/2-inch loaf pan with butter.
Sift flour, baking powder, and salt together into a bowl.
Combine white sugar and lemon zest in a bowl. Rub lemon zest into the sugar with your fingers until fragrant.
Combine eggs, yogurt, juice of 1 lemon, and vanilla extract in a large bowl; beat with an electric mixture until blended. Add flour mixture and sugar mixture; beat on low speed until batter is just combined. Fold in olive oil gently using a spatula.
Pour batter into the prepared loaf pan.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, 40 to 50 minutes. Let cool in the pan, at least 30 minutes. Run a knife around the edge of the pan to loosen; invert onto a serving dish.
Whisk remaining lemon juice and confectioners' sugar together in a bowl to make a thin glaze. Drizzle glaze over cake.