Preheat oven to 350°F. Grease a 9x13-inch baking pan with butter and lightly flour it, tapping out excess. Set aside.
In a large mixing bowl, use an electric mixer on high speed to beat the eggs for 5 minutes until thick, pale yellow, and fluffy (they should triple in volume). Add sugar and vanilla. Beat on medium-high for 2-3 minutes until well combined and creamy.
In a separate bowl, whisk together flour, baking powder, and salt. Set aside.
In a small saucepan over medium heat, combine milk and butter. Heat, stirring occasionally, until butter melts and small bubbles form around edges (about 170°F—hot but not boiling). Remove from heat immediately.
With mixer on low speed, gradually add dry ingredients to egg mixture in two batches, mixing just until no dry streaks remain (don't overmix). Slowly pour in hot milk mixture in a steady stream while mixing on low. Beat 30-60 seconds until batter is smooth and thin. Scrape bowl with spatula.
Pour batter into prepared pan. Tap pan gently on counter to release air bubbles. Bake 35-40 minutes until top is golden, cake springs back when lightly touched, and toothpick inserted in center comes out clean or with a few moist crumbs. Cool in pan on wire rack 10 minutes, then cool completely.