Preheat the oven to 350 degrees F. Grease and flour two 8-inch cake pans. I also lined the bottom of my cake pans with rounds of parchment.
In a large bowl, mix together the flour, baking powder and salt.
To the bowl of a stand-up mixer, with the paddle attachment (alternatively, you could use an electric hand mixer), add the lemon zest and sugar. Mix on low until the lemon zest is mixed throughout the sugar (this really brings out the flavor in the lemon zest!). Add the butter and beat until fluffy, about 1 minute. Crack in the eggs and add the lemon juice and vanilla extract. Mix until completely incorporated. Add the flour mixture and beat just until the ingredients are combined.
Divide the batter between the two cake pans. Transfer to the oven to bake for 15 to 18 minutes, until a skewer inserted into the center comes out clean. Allow to cool in the pans for about 5 minutes, until inverting them onto cooling racks.