Preheat oven to 350 degrees. Butter and flour two 8-by-2-inch cake pans, tapping out excess flour. In a medium bowl, whisk flour, baking powder, baking soda, salt, and lemon zest.

In a large bowl, using an electric mixer, beat butter and 1 1/2 cups sugar until light and fluffy. With mixer on low, beat in eggs and yolks, one at a time. Beat in 2 tablespoons lemon juice. Alternately beat in flour mixture and buttermilk beginning and ending with flour mixture; mix just until combined.
Divide batter between pans; smooth tops. Bake until cakes pull away from sides of pans, 32 to 35 minutes. Let cool in pans 10 minutes. Run a knife around edges of pans and invert cakes onto a wire rack.

While cakes are baking, bring remaining 1/2 cup sugar and 1/2 cup water to a boil in a saucepan. Add lemon slices and simmer 25 minutes. Using a slotted spoon, transfer lemon slices to a waxed-paper-lined plate. Stir remaining 1/4 cup fresh lemon juice into syrup.

Using a toothpick, poke holes in warm cakes on rack. Brush with lemon syrup. Let cool completely. Prepare frosting, substituting 2 tablespoons fresh lemon juice for vanilla extract.