Preheat the oven to 350 F (180 C) at least 20 minutes before. Line a 7" cake pan with parchment paper (bottom and side). Put butter and milk In a small bowl and microwave until the butter melts. In another bowl, sift the cake flour 2 times. Set aside.
In a bowl of a stand mixer, add eggs and sugar. Over a double boiler (placing the bowl over a pot with simmering water on low heat), whisk the egg mixture until it becomes lukewarm (until the sugar dissolves). Then set the bowl on the stand mixer, and whip the egg mixture until very thick (watch video to see the texture). Add vanilla and mix.
Add sifted cake flour in 3 parts. Fold in the first one-third of the flour by hand with a spatula (not with the mixer). After the flour is incorporated, add the next part of flour and again fold in with the spatula. Repeat. Next, add the mixture of melted butter and milk and fold in with the spatula.
Put the batter in the prepared pan. Bake for 20-25 minutes.