Make sure to really cream the butter and sugar well, and use whole fat milk if possible! This makes for a more rich flavor of cake.
Preheat oven to 350°F. Line the bottom of each cake pan with parchment paper.
Sift the flour, baking powder, and salt into a medium bowl and whisk together. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, beat the butter, sugar and vanilla on medium speed until light and fluffy, about 8 minutes.
Add the eggs, two at a time, beating until each addition is incorporated before adding the next. Scrape down the sides of the bowl with a spatula when necessary.
Add the flour mixture in four parts, alternating with the milk in three parts, beating just until each addition is incorporated before adding the next; do not overmix.
Scrape the batter into the prepared pan or pans and spread it so that it is smooth in the pan(s). Bake until a toothpick inserted in the center comes out clean, about 40 minutes for the 9 x 13-inch pan or minutes for 45 minutes for the 8-inch pans, rotating the pan(s) halfway through.
Transfer to a wire rack and let cool completely in the pan(s). Cover tightly with plastic wrap and refrigerate overnight. Loosen the edges with a straight spatula, invert the pan(s) to remove the cake, and peel off the parchment.