Butter the bottoms and sides of two 9 inch round cake pans; line the bottoms with parchment paper. Preheat the oven to 350°.
Combine milk and 4 tbsp of unsalted butter in a small saucepan. Heat over low heat until the butter is just melted. Set the pan aside.
Stir flour and baking powder together, then sift onto a piece of parchment paper.
Whisk the eggs in a mixing bowl by hand to combine, and then whisk in the salt. Whisk in sugar and vanilla extract. Then, using a hand held mixer, whip the mixture on medium-high speed until lighter in color and very light and fluffy in texture (about 5 minutes).
Gently whisk in the warm milk mixture by hand. Add in the flour mixture in 4 additions, lifting the parchment paper to help you sprinkle it in. Use the whisk to fold the flour into the liquid, gently whisking to incorporate each time. Divide the batter between the prepared pans and smooth tops.
Bake in the center of the oven until golden on top and a cake tester comes out clean (15-17 minutes). Let cool in the pans on wire racks for about 10 minutes.
Run a sharp knife around the layers to loosen, then invert the pans onto the racks.
Immediately turn the layers over so they cool with the paper on the bottom. Let cool completely.