Preheat the oven to 350 degrees F. In a small saucepan, warm the milk and 2 teaspoons of the butter together over medium-low heat. With an electric mixer fitted with a wire whip, beat the eggs and 1 cup of the sugar on medium-high speed in a large mixing bowl until the mixture is pale yellow, thick, and tripled in volume, about 8 minutes. With the mixer on low speed, beat in the warm milk mixture. Sift the flour, baking powder, and salt into a small mixing bowl. Add half the flour mixture to the egg mixture and blend thoroughly until smooth. Repeat with the other half. Add the vanilla and mix gently. Grease a 17 by 12-inch baking pan or jelly-roll pan with the remaining 2 tablespoons butter. Sprinkle evenly with the remaining 2 tablespoons sugar. Pour the cake batter into the pan, spreading it evenly. Bake until the cake springs back when touched, about 15 minutes. Cool for about 2 minutes, then gently flip it out onto a large sheet of parchment paper. Let cool completely.