Preheat the oven to 350 °F.
Butter and lightly flour two 9-inch round baking pans and line the bottom with parchment paper. I used springform pans as that makes removing the cakes a bit easier.
Sift together the flour, baking powder and salt. Set aside.
Heat the milk and butter in a small pan over medium-low heat just until small bubbles form around the edge of the pan. Do not allow this mixture to boil. Off the heat, add the vanilla extract. Cover the pan and set it aside.
Combine the eggs and sugar in a bowl of an electric mixer fitted with a paddle attachment. Beat on medium speed until the mixture is pale yellow and thick (about 4-5 minutes). You'll know that the mixture is ready when it falls back on itself in a ribbon and lingers on the surface for about 10 seconds.
By hand, gently whisk in the warm milk mixture. In three separate additions, slowly whisk in the flour mixture until just blended. Be careful not to over-mix.
Divide the mixture between the two prepared pans. Bake for about 25 minutes or until the cakes are lightly brown and a toothpick inserted in the center comes out clean. Allow the cakes to cool for about 15-20 minutes then remove them from the pans and allow them to cool completely on a baking rack.