Preheat oven to 350°F with a rack in the center. Butter and flour a 9 x 13 inch baking pan, tapping out any excess flour.
Into a medium mixing bowl, sift together the flour, baking powder, and salt.
In a large mixing bowl or the bowl of a stand mixer fitted with the whisk, beat butter and sugar together on medium-high until pale and fluffy, about 5 minutes. Beat in the eggs one at a time until incorporated. Add vanilla and beat for 5 minutes more.
Reduce mixer speed to low and alternate between adding the flour mixture (in 3 parts) and the milk (in two parts), starting and ending with flour. Mix just until batter is smooth — do not overmix.
Spread batter into pan and bake for about 20 minutes (check at 18), just until a cake tester comes out clean from the center. Cake will be pale, not brown. Cool completely before frosting.