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Put the vanilla seeds in a large bowl. Add the egg yolks and sugar syrup, and beat with an electric whisk until pale, about 2-3 mins. Add the cream, milk and brandy, and beat again until just combined. Strain into a punch bowl.
Clean the beaters, then beat the egg whites in a clean bowl to soft peaks. Fold a third of the whipped egg whites into the milk and cream mixture in the punch bowl, then add the remaining whipped egg whites. Gently fold these in until the eggnog is fluffy and smooth. Chill until ready to serve. Will keep, covered in the fridge, for up to a day. Just before serving, add plenty of ice, then stir gently and ladle into tumblers.