Preheat oven to 375°F. Generously butter a 9-inch square baking pan (at least 2 inches deep) or a 9-inch round cake pan (at least 2 inches deep).

Beat sugar and butter in bowl of a stand mixer fitted a paddle attachment on high speed until light and fluffy, about 5 minutes. Add eggs, 1 at a time, beating well after each addition. Beat in vanilla.

Whisk together flour, baking powder, nutmeg, and salt in a medium bowl. Add to butter mixture in 3 additions, alternating with one-third of the milk, beginning and ending with flour mixture. Beat on low speed until batter is smooth after each addition, scraping bowl to ensure all of the flour mixture is incorporated.

Pour batter into prepared pan, and smooth top. Bake in preheated oven until a wooden pick inserted in center comes out clean, 30 to 35 minutes. If the top browns too quickly before the cake is done, cover top with a sheet of aluminum foil. Cool in pan on a wire rack 10 minutes. Remove cake from pan to wire rack, cut into squares, and serve warm.