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Preheat oven to 350° F. Grease and parchment line two 9” pans, then grease again. In a medium bowl combine eggs, 1/4 cup milk and vanilla. In a large bowl combine dry ingredients. Slowly combine into the dry ingredients butter and 3/4 cup milk. Once ingredients are moistened, increase to medium mixing speed and beat 1 1/2 minutes to aerate batter. Scrape down the batter and add the egg/milk mixture to the cake batter in 3 batches, scraping down the sides between each addition. Pour into prepared pans and bake 25-30 minutes, until top springs back lightly when touched and before the edges pull from the pan. FOR THE FROSTING: whisk together flour and sugar while scalding the milk. Whisk milk into dry mixture in a pan and cook over a medium high flame, constantly stirring, until mixture thickens and begins to bubble. Pour into a mixing bowl and let cool. Once cool, beat in butter and chestnut puree. Beat until smooth and fluffy, and frost the cake.