Preheat oven to 350 degrees F. Line two 9-inch round cake pans or one 9x13-inch rectangular baking pan with parchment paper. Grease the paper and the sides of the pan well.

In a large mixing bowl, with an electric mixer, beat sugar and eggs together until slightly thickened, about 1 minute. Add flour, milk, oil, baking powder, and vanilla and beat for another minute, just until the batter is smooth and creamy. Don't overbeat. Pour batter into the prepared baking pan(s).

Bake in preheated oven for 30 to 40 minutes or until the tops are golden and a toothpick poked into the center of the layer comes out clean. (A single rectangular pan will take longer to bake than two round ones.) Loosen the sides of the cake from the pan with a thin knife, then turn out onto a rack and peel off the paper. Let cool completely before covering with frosting.