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Pre-heat the oven to 350˚F and grease a 9” x 13” baking pan.
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Use a hand mixer or stand mixer to beat the eggs for about five minutes, until they are thick and light in color. Add the sugar to the eggs gradually, continuing to beat with each addition. Beat in the vanilla extract.
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In a second bowl, combine the flour, baking powder and salt. Add this to the eggs and mix in on low speed until you can no longer see any streaks of white.
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Combine the milk and butter in a saucepan and scald the milk – this means, bring the mixture to a temperature where the butter has melted and you start to see little bubbles around the side of the pan. If you have an instant read thermometer, you are looking for 170˚F. Remove the milk and butter from the heat and gradually add it to the cake batter, beating it in gently. Do not overmix.
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Transfer the batter to the greased baking pan and send it to the oven. Bake for 35 minutes, or until a toothpick inserted into the center of the cake comes out clean.
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Remove the pan from the oven and let the cake cool for 10 minutes. Then remove it from the pan and let it cool completely on a cooling rack.
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To make the brown sugar icing: Combine the brown sugar, granulated sugar and milk in a saucepan. Bring the mixture to a boil until it reaches soft ball stage (235˚F). (If you don’t have an instant read thermometer, you can tell you’ve reached the right temperature by dropping a drop of the mixture into a glass of ice water. If the brown sugar forms a little ball that you can squish between your fingers, you’ve reached soft ball stage.) Whisk in the butter and vanilla let the mixture cool. Beat in the powdered sugar until the icing reaches the consistency you like for spreading on top.
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Ice the cake with the brown sugar icing and serve. Store leftovers well covered, at room temperature for 3 to 4 days.