Preheat oven to 350 degrees F (175 degrees C). Grease and flour a 9×13-inch baking pan.
Beat eggs in a stand mixer fitted with the whisk attachment on high speed until thick and lemon-colored, about 3 minutes. Gradually add sugar, beating until mixture is thick, about 5 minutes. Fold flour, baking powder, and salt into the batter by hand.
Microwave milk in a small microwave-safe bowl until it begins to bubble, about 1 minute. Stir in butter until melted. Add vanilla extract.
Pour milk mixture into the batter; blend on low speed until well-combined and smooth. Pour into the prepared pan.
Bake in the preheated oven until a toothpick inserted in the center comes out clean, 25 to 30 minutes.