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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 flour, baking powder, and salt into the batter by hand.
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 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.