Step 1Heat the oven to 325 degrees.
Step 2In a large bowl, sift together the cake flour, 1 1/4 cups sugar, baking powder and salt. Whisk the dry ingredients well to make sure they are thoroughly combined. Make a well in the center of the dry ingredients.
Step 3In a small bowl, beat the egg yolks. Pour the egg yolks into the well, along with the vegetable oil, milk and vanilla. Beat the wet ingredients into the dry until completely smooth.
Step 4In the bowl of a stand mixer or in a large bowl using a hand mixer, beat the egg whites and cream of tartar until foamy. With the mixer running, slowly rain in the remaining one-fourth cup sugar. Continue to beat the whites until stiff peaks form when the beater is lifted.
Step 5Fold the beaten whites into the rest of the batter: Gently spoon one-third of the beaten egg whites into the large bowl with the batter. Slowly and carefully fold the whites into the batter using a spatula or whisk until mixed. Add another third of the beaten whites to the bowl and gently fold into the batter. Be very gentle as you fold in the whites as you do not want to deflate them; the whites lighten the batter and are largely responsible for the cake's ability to rise as it bakes. Gently fold in the remaining third of the whites.
Step 6Spoon or gently pour the batter into a 10-inch ungreased angel food cake pan. Place the pan in the oven and bake until the cake is puffed (it should rise over the top of the pan by 2 to 3 inches but will deflate a little as it cools), lightly browned on top and a toothpick or cake tester inserted comes out clean, 1 hour to 1 hour and 10 minutes.
Step 7Remove from heat and invert the pan over a wine or soda bottle. Set the pan aside until cooled completely, 1 to 2 hours.
Step 8Loosen the sides with a thin knife or metal spatula and tap it gently to remove the cake (if using a two-piece pan, loosen the outside of the pan to remove, then gently work the knife or spatula along the top of the insert and inside to remove the cake) before serving.