3 cups sifted all-purpose flour, plus more for dusting
1 tablespoon baking powder
3/4 teaspoon salt
1 cup milk, at room temperature
2 teaspoons pure vanilla extract
1 1/2 sticks unsalted butter, softened
2 cups granulated sugar
4 large eggs, at room temperature
2 teaspoons finely grated lemon zest
1/2 pound unsalted butter, softened
1 large egg yolk (optional)
1 pound confectioners' sugar
1/4 cup plus 2 tablespoons heavy cream
1 teaspoon pure vanilla extract
Pinch of salt
1 teaspoon finely grated lemon zest
1 1/2 tablespoons fresh lemon juice
Preheat the oven to 350°. Butter two 9-by-2-inch round cake pans and line with parchment paper; butter and dust with flour. In a bowl, mix the 3 cups of flour, baking powder, and salt. In a cup, mix the milk with the vanilla.
In a standing mixer fitted with the paddle, beat the butter at medium speed until light and creamy. Add the sugar and beat until fluffy, 4 minutes. Beat in the eggs, one at a time, scraping down the bowl. Beat in the dry ingredients in 3 batches, alternating with the milk mixture and scraping down the bowl. Beat in the grated lemon zest.
Scrape the batter into the pans. Bake in the lower third of the oven for about 35 minutes, until springy. Let cool for 15 minutes, then run a knife around the edges and invert onto a rack. Peel off the paper, turn the cakes upright and cool completely.
In a standing mixer fitted with the whisk, beat the butter and egg yolk at medium-high speed until creamy. Beat in the sugar at low speed. Beat in the cream, vanilla and salt, then beat at medium-high speed until fluffy, 3 minutes longer. Beat in the lemon zest and juice.
Spread 1 cup of frosting over a cake layer and top with the second layer; frost the top and sides. Let stand 1 hour; serve.