Preheat the oven to 350 degrees F (175 degrees C). Grease and flour three 8- or 9-inch-diameter cake pans.

Cream sugar and butter together in a large bowl with an electric mixer until smooth and fluffy, about 5 minutes. Add egg yolks, one at a time, beating well after each addition. Mix in vanilla.

Sift flour, baking powder, and a pinch of salt together into a bowl. Add flour mixture to butter mixture in three batches, alternating with milk, beating after each addition until just combined. Scrape down the sides and bottom of the bowl and beat for 1 minute more.

Beat egg whites with remaining pinch of salt in a separate bowl until stiff peaks form. Fold 1/3 of the egg whites into cake batter to lighten it; then gently fold in remaining egg whites. Pour batter evenly into the prepared pans.

Bake in the preheated oven until cakes spring back when lightly touched with a fingertip, and a tester inserted into the center comes out clean, about 25 minutes. Remove from the oven and cool on a wire rack.