Preheat oven to 350 degrees F. Butter a 9x9-inch baking pan.

For cake: In a saucepan over low heat, warm the milk until it is hot to the touch. Remove from the heat, add butter, and set aside.

Combine eggs and sugar in a bowl. Using an electric mixer, beat on high speed for 3 minutes, or until light and airy. Add vanilla and beat to blend.

In a separate bowl, combine the flour, baking powder, and salt.

Add flour mixtures, about one-third at a time, to the egg mixture, beating on low speed after each addition. Add milk and butter and beat on medium-low speed for about 30 seconds, or until blended. Pour into the prepared pan.

Bake on center oven rack for 30 to 35 minutes, or until the top springs back when touched and a toothpick inserted into the center comes out clean.

Transfer to a cooking rack for 20 to 30 minutes.