Cut butter into 16 pieces and place in the bowl of an electric mixer; let stand at room temperature for 30 minutes to soften slightly. In a small bowl, use a fork to beat together the eggs, yolks and vanilla extract. Let stand at room temperature until ready to use.
Preheat oven to 325 degrees F. Grease and flour an 8½ by 4½-inch loaf pan.
Beat the butter and salt at medium-high speed until smooth and creamy, about 3 minutes, stopping to scrape the bowl once. Reduce the mixer speed to medium and gradually pour in the sugar. Once all of the sugar is added, increase the speed to medium-high and beat until light and fluffy, 5 to 8 minutes, scraping the bowl once. Reduce the mixer speed to medium and gradually add the egg mixture in a slow, steady stream. Scrape down the bowl, then beat the mixture at medium-high speed until light and fluffy, 3 to 4 minutes. Scrape down the bowl.
Sift the flour over the batter in 3 additions, folding gently with a rubber spatula after each addition.
Pour the batter into the prepared pan and smooth the top. Bake until golden brown and a toothpick inserted in the center comes out clean, 1 hour 10 minutes to 1 hour 20 minutes, rotating pan halfway through baking. Let cake cool in the pan on a wire rack for 15 minutes, then turn out onto a wire rack and cool completely, at least 2 hours, before serving. The cake can be stored, wrapped tightly in plastic wrap, at room temperature for up to 3 days.