Step 1

Preheat the oven to 350 degrees F. Grease an 8 1/2 by 4 1/4 by 2 1/2-inch loaf pan. Line the bottom with parchment paper. Grease and flour the pan.

Step 2

Sift together the flour, baking powder, and salt into 1 bowl. In another bowl, whisk together the yogurt, 1 cup sugar, the eggs, lemon zest, and vanilla. Slowly whisk the dry ingredients into the wet ingredients. With a rubber spatula, fold the vegetable oil into the batter, making sure it's all incorporated. Pour the batter into the prepared pan and bake for about 50 minutes, or until a cake tester placed in the center of the loaf comes out clean.

Step 3

Meanwhile, cook the 1/3 cup lemon juice and remaining 1/3 cup sugar in a small pan until the sugar dissolves and the mixture is clear. Set aside.

Step 4

When the cake is done, allow it to cool in the pan for 10 minutes. Carefully place on a baking rack over a sheet pan. While the cake is still warm, pour the lemon-sugar mixture over the cake and allow it to soak in. Cool.

Step 5

For the glaze, combine the confectioners' sugar and lemon juice and pour over the cake.