Butter the bottom and sides of two 9-inch round cake pans; line the bottoms with parchment paper. Preheat oven to 350°F.
Combine milk and 4 tablespoons 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 eggs in a mixing bowl by hand to combine, then whisk in salt. Whisk in sugar in a stream, then whisk in vanilla. Then, using a hand-held electric mixer or a stand mixer with the whisk attachment, whip the mixture on medium-high speed until lighter in color and very light and fluffy in texture, about 3 minutes in a stand mixer or 5 minutes with a hand mixer.
Gently whisk in the warm milk mixture by hand. Add the flour mixture in 4 additions, lifting the parchment paper to help you sprinkle it in. Use the whisk as you would a rubber spatula to fold the flour into the liquid, gently whisking to incorporate each time. Divide the batter between the prepared pans and smooth the tops.
Bake in the center of the oven until golden and firm when touched in the center, about 20 minutes. Let cool in the pans on a wire racks for 10 minutes.
Run a sharp paring 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.
Wrap cooled cake airtight and refrigerate for up to 5 days or freeze for up to 3 months. Defrost at room temperature.
To bring eggs to room temperature, either set them out on the counter for 15 minutes or submerge them (in the shell) in a bowl of lukewarm (not hot) water for 5 minutes; the eggs will beat to a greater volume.
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