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Preheat the oven to 170°C (150°C fan) | 325F | gas 3. Place one 1 lb jam jar in the oven with its lid to sterilise for 10 minutes.
Place the cooked chestnuts in a saucepan and cover with the milk and vanilla extract. Simmer for 10 minutes, then drain and transfer the chestnuts to a food processor.
Add the chestnut puree and some of the poaching milk and pulse the mixture until it comes together to form a smooth jam. Add a little more milk if you need to loosen the jam.
Remove the jar from the oven and fill with the jam. Seal well and let it cool to room temperature before serving with sprigs of mint and slices of baguette.