Preheat oven to 350°F (175°C). Grease and flour three 8-inch round cake pans.

In a large bowl, whisk together flour, sugar, baking powder, and salt.

Add butter and milk to the dry ingredients and beat until just combined.

Add eggs, lemon zest, and vanilla extract and beat until smooth.

Pour batter evenly into prepared pans.

Bake for 30-35 minutes, or until a wooden skewer inserted into the center comes out clean.

Let cakes cool in pans for 10 minutes before inverting onto a wire rack to cool completely.

In a large bowl, beat butter until light and fluffy.

Gradually add powdered sugar, beating until smooth.

Add lemon juice, lemon zest, and vanilla extract and beat until well combined.

Place one cake layer on a serving plate and spread with frosting.

Repeat with remaining cake layers and frosting.

Frost the top and sides of the cake with remaining frosting.