Preheat the oven to 350 degrees F (175 degrees C). Grease and flour a 9x13-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 in flour, baking powder, and salt by hand until just incorporated.

Microwave milk in a microwave-safe bowl until it begins to bubble, about 1 minute. Add butter and stir until melted. Stir in vanilla.

Pour hot milk mixture into egg mixture; blend on low speed until batter is well combined and smooth. Pour into the prepared pan.

Bake in the preheated oven until a toothpick inserted into the center comes out clean, 25 to 30 minutes.