Adjust the oven rack to the middle position and heat the oven to 350 degrees. Grease and flour a 9 x 13, broiler safe baking pan (Do not use Pyrex). Whisk together the flour, baking powder, and salt in a large bowl. Set aside. Heat the butter and the milk in a small saucepan over medium heat until melted. Stir in the vanilla.
In a medium bowl, beat together the eggs and granulated sugar with an electric mixer on medium high until pale and thick, about 6 minutes. Fold half of the flour mixture into the egg mixture, then stir in 1/2 of the milk mixture, stir the other half of the flour mixture into the egg mixture, then stir in the other 1/2 of the milk mixture.
Pour the batter into the prepared pan and bake until a toothpick inserted in the center of the cake comes out clean, 30-35 minutes. Let cool slightly in the pan, about 10 minutes. Heat the broiler.