Preheat the oven to 350 degrees F.
Grease the bottom and sides of a 9-inch cake pan.
Whisk together the cake flour, baking powder, and salt. Set aside.
In a small saucepan, combine the milk and butter.
Heat over medium heat just until the butter is melted. Remove the pan from the heat and set aside.
In a bowl of an electric mixer using the whisk attachment, beat the eggs at high speed until blended, about 1 minute. Gradually add the sugar and vanilla extract and beat until pale and tripled in volume, about 6 minutes.
Gently fold one-third of the flour mixture into the egg mixture. Repeat with the remaining flour mixture in two more additions.
Reheat the milk mixture to just under a boil.
Add it all at once to the egg mixture and gently fold it in. Scrape the batter into the prepared pan.
Bake the cake for 20-25 minutes, until it springs back when lightly touched and a cake tester inserted into the center comes out clean.
Cool the cake in the pan on a wire rack for 10 minutes. Run a paring knife around the edge of the pan and invert the cake onto the wire rack.