Step 1

Adjust oven rack to middle position and heat oven to 350 degrees. Grease two 9-inch or three 8-inch round cake pans, line with parchment paper, grease parchment, and flour pans.

Step 2

Mix milk, egg whites, and vanilla together in 2-cup liquid measuring cup. Using stand mixer fitted with paddle, mix flour, sugar, baking powder, and salt on low speed until combined, about 5 seconds. Add butter, 1 piece at a time, and mix until only pea-size pieces remain, about 1 minute.

Step 3

Add half of milk mixture, increase speed to medium-high, and beat until light and fluffy, about 30 seconds. Reduce speed to medium-low, add remaining milk mixture, and beat until incorporated, about 15 seconds (batter may look slightly curdled). Give batter final stir by hand; do not overmix.

Step 4

Divide batter evenly between prepared pans and smooth tops with rubber spatula. Bake until tops are light golden and toothpick inserted in center comes out clean, 18 to 22 minutes (for 8-inch pans) or 23 to 25 minutes (for 9-inch pans), switching and rotating pans halfway through baking. Let cakes cool in pans on wire rack for 10 minutes. Remove cakes from pans, discarding parchment, and let cool completely on rack, about 2 hours. (Once cooled, cake layers can be wrapped and stored for one day.)