Preheat the oven to 350 degrees, butter three 8-inch cake pans then line the bottoms with parchment paper. Butter the parchment and flour the pans to get rid of the excess.

In a bowl, mix 1/4 cup of the milk with the egg whites and vanilla extract. In the bowl of a standing mixer fitted with the paddle, mix the flour with the sugar, baking powder and salt. Add the butter and the remaining 3/4 cup of milk. Beat at low speed until blended, then beat at medium speed until smooth, 1 minute. Beat in the egg white mixture in 3 batches.

Get another bowl, use a electric mixer and beat the cream until soft peaks form. Stir 1/3rd of the whipped cream into the batter and fold in the rest. Divide the batter between the pans and smooth the tops. Bake it for 25 minutes or until a toothpick is put in the middle and it comes out with nothing on it. Let the cake cool on a rack for about 10 minutes, then unmold the cakes and peel off the parchment. Invert the cakes and let cool completely.

In a saucepan, stir 2 1/2 cups of the sugar with the corn syrup and milk and cook over moderate heat, stirring until the sugar dissolves. Make sure it stays warm.

Sprinkle the remaining 1/2 cup of sugar in a deep and heavy saucepan. Cook the sugar with moderate heat then stire it occasionally, until an amber caramel forms. Carefully pour the warm milk mixture over the caramel. Cook over moderately high heat and stir it until the caramel dissolves. Stop stirring and cook until the caramel it says 235 degrees on a candy thermometer then remove from the heat and stir in the butter, vanilla and 1/4 cup of the heavy cream. Put the caramel into the bowl of a standing mixer and let cool for 15 minutes.

Beat the caramel at a steady speed, gradually adding the remaining 1/4 cup of cream, until creamy, for about 15 minutes.

Set 1 cake layer on a plate and pour enough icing over the layer to cover the top. Top with a second cake layer and cover it with icing, add the final cake layer and pour the rest of the icing over the top of the cake and let it run down the sides. Working quickly, use an offset spatula to spread the icing gently around the cake. Let the cake stand for 2 hours to set the icing before serving.