Position a rack in the center of the oven. Preheat the oven to 350°F (325°F for a convection oven). Grease three 9- by 1½-inch round baking pans or two 9- by 3-inch round baking pans or springform pans, line the bottoms with parchment paper or wax paper, grease again, and dust with flour.To make the batter: Sift together the flour, baking powder, salt, and, if using, nutmeg.
In a large bowl, beat the butter on low speed until smooth, about 2 minutes.
Increase the speed to medium, gradually add the sugar, and beat until light and fluffy, about 5 minutes. Add the vanilla.Add the flour mixture and milk alternately (4 portions for the flour; 3 portions for the milk) beginning and ending with the flour.In a large bowl, beat the egg whites on low until foamy, about 1 minute. Increase the speed to medium and beat until soft leaks form, 5 to 8 minutes. Fold one fourth of the egg whites into the batter, then gently fold in the remaining whites.
Divide the batter equally between the prepared pans. Bake until a tester inserted in the center comes out clean and the cake springs back when lightly touched, 20 to 25 minutes for 3 pans or about 30 minutes for 2 pans.
Let cool in the pans for 10 minutes, then remove to wire racks and let cool completely, at least 1½ hours. The cake can be wrapped and stored at room temperature for up to 1 day.