Heat the oven to 180°C/gas 4. Grease an ovenproof dish with a little of the butter. Warm the remaining butter and the milk in a saucepan. Cut the vanilla pod in half and scrape the seeds out of the pod into the milk. Place the polenta, flour and icing sugar in a mixing bowl. Mix together and make a well in the centre.
Crack the eggs into a bowl and whisk. Stir in the warm milk, then pour into the polenta and flour mixture. Mix until smooth. Add the lemon zest, juice and Armagnac. Stir well and pour into the greased dish. Bake in the oven for 25 minutes.