Cut butter into tablespoon pieces and place in bowl of standing mixer; let stand at room temp for 20-30 min to soften. Using dinner fork, beat eggs, yolks, and vanilla in liquid measuring cup until combined. Let egg mixture stand at room temp until ready to use.
Adjust over rack to middle and heat to 325 degrees. Generously butter 9x5 inch loaf pan; dust with flour and knock out excess. Alternatively, spray pan with cooking spray and line with parchment.
In standing mixer fitted with flat beater, beat butter and salt at medium-high speed until shiny, smooth and creamy, 2-3 minutes, scraping bottom and sides of bowl once with rubber spatula. Reduce speed to medium; with mixer running, gradually pour in sugar (should take about a minute total). Once all sugar is added, increase to medium-high and beat until mixture is fluffy and almost white in color, 5-8 minutes, scraping bottom and sides of bowl once. With mixer running at medium speed, gradually add egg mixture in slow, steady stream; this should take 60-90 seconds. Scrape bottom and sides of bowl; beat mixture at medium-high speed until light and fluffy, 3-4 minutes (mixture may look slightly broken). Remove bowl from mixer; scrape bottom and sides.
In 3 additions, add flour over butter/egg mixture, after each addition, fold gently with rubber spatula until combined. Scrape along bottom of bowl to ensure that batter is homogeneous.
Transfer batter to prepared pan and smooth surface with rubber spatula. Bake until golden brown and wooden skewer inserted into center of cake comes out clean, about 70-80 minutes. Cool cake in pan on wire rack for 15 minutes, invert cake onto wire rack, then turn cake right side up. Cool cake on rack to room temp, about 2 hours. Slice and serve.