Preheat oven to 350 degrees. Grease a 8 1/2 x 4 1/4 x 2 12-inch loaf pan with baking spray. Line bottom of pan with parchment paper. Grease and flour with baking spray and flour.
Sift together flour, baking powder,salt in a medium bowl.
In another medium 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, until a cake tester placed in the center of the loaf comes out clean. (For this you could use a chop stick or the probe end of a cooking thermometer. I start checking at 45 minutes and keep adding time in five minute incremements until the tester is clean.)
While the cake is baking, cook 1/3 cup of sugar with 1/3 cup of lemon juice in a small pan over low heat until the sugar dissolves and the mixture is clear. Set aside.
When the cake is done, remove from the oven and let it cool in the pan for 10 minutes. Then, carefully remove from the pan and place it on a baking rack over a sheet pan to cool. Now, take the chop stick or thermometer probe and poke holes all over the cake. While the cake is still warm, pour the lemon-sugar mixture over the cake and allow it to soak in. Cool completely.
For the glaze, sift and combine the confectioners' sugar and lemon juice until smooth (add extra lemon juice until the icing is pourable. Pour over the cake.
Enjoy!