Preheat the oven to 350 degrees. Butter three 8-inch cake pans; line the bottoms with parchment paper. Butter the parchment and flour the pans, tapping out 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.

In another bowl, using an electric mixer, beat the cream until soft peaks form. Stir one-third of the whipped cream into the batter, then fold in the rest. Divide the batter between the pans and smooth the tops. Bake for 25 minutes, until a toothpick inserted in the centers comes out clean. Let the cakes cool on a rack for 10 minutes. 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. Cook over moderate heat, stirring until the sugar dissolves. Keep warm.

Sprinkle the remaining 1/2 cup of sugar in a deep, heavy saucepan. Cook the sugar over moderate heat, swirling occasionally, until an amber caramel forms. Carefully pour the warm milk mixture over the caramel. Cook over moderately high heat, stirring, until the caramel dissolves. Stop stirring and cook until the caramel registers 235 degrees on a candy thermometer. Remove from the heat. Stir in the butter, vanilla and 1/4 cup of the heavy cream. Strain the caramel into the bowl of a standing mixer. Let cool for 15 minutes.

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

Set 1 cake layer on a plate. 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, letting 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.