Set pureed carrots aside in a large mixing bowl to cool.
Melt butter in a heavy saucepan over low heat; add flour, stirring until smooth. Cook 1 minute, stirring constantly. Gradually add milk; cook over medium heat, stirring constantly, until mixture is thickened and bubbly.
Beat eggs in a small mixing bowl; gradually stir in one fourth of hot white sauce. Add egg mixture to remaining white sauce, stirring constantly. Remove from heat. Add white sauce, sugar, nutmeg, salt, and vanilla to reserved carrots; mix well. Spoon mixture into a 2- quart soufflé dish. Bake at 350° for 1 hour and 25 minutes. Remove from oven, and let stand 5 minutes before serving.